diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdqfe" "b/data_all_eng_slimpj/shuffled/split2/finalzzdqfe" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdqfe" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\nImage quality has become a critical evaluation metric in most image-processing applications, including image denoising, image super-resolution, compression artifacts reduction, \\etc. Directly acquiring perceptual quality scores from human observers is accurate. However, this requires time-consuming and costly subjective experiments. The goal of Image Quality Assessment (IQA) is to allow computers to simulate the Human Visual System (HVS) through algorithms to score the perceptual quality of images. In this case, the images to be evaluated are often degraded during compression, acquisition, and post-processing. \n\nIn recent years, the invention of Generative Adversarial Networks (GANs)~\\cite{goodfellow2014generative} has greatly improved the image processing ability, especially image generation~\\cite{gu2020image,xia2021tedigan} and image restoration~\\cite{wang2018esrgan}, while it also brings new challenges to image quality assessment. GAN-based methods can fabricate seemingly realistic but fake details and textures~\\cite{jinjin2020pipal}. In detail, it is hard for the HVS to distinguish the misalignment of the edges and texture decreases in the region with dense textures. As long as the semantics of textures are similar, the HVS will ignore part of the subtle differences of textures. Most IQA methods for traditional distortion images assess image quality through pixel-wise comparison, which will lead to underestimation for GAN-generated images~\\cite{wang2004image}. To deal with the texture misalignment, recent studies~\\cite{bosse2017deep} introduce patch-wise prediction methods. \nSome following studies~\\cite{shi2021region,jinjin2020pipal} further propose different spatially robust comparison operations into the CNN-based IQA network. However, they take each patch as an independent input and separately calculate their score and weight, which will lead to the loss of context information and the inability to model the relationship between patches. \n\nTherefore, on the basis of patch-level comparison, we need to better model the interrelationship between patches. To this end, we use Vision Transformer (ViT)~\\cite{dosovitskiy2020image} as a feature extractor, which can effectively capture long-range dependencies among patches through a multi-head attention mechanism. However, the vanilla ViT uses a large convolution kernel to down-sample the input images in spatial dimension before entering the network; some details that should be considered are lost, which are also crucial to image quality assessment. Based on the observation, we found that a shallow CNN is a good choice to provide detailed spatial information. \nThe features extracted by a shallow CNN contains unwanted noises and merging ViT features with them would decrease the performance. To alleviate the impact of noise, we propose to mimic the characteristic of the HVS that human always pay attention to the salient regions of images. \nInstead of injecting the complete features from a shallow CNN into those from ViT, we only use those that convey spatial details of the salient regions for image quality assessment, thereby alleviating the aforementioned noise.\nFurthermore, using max-pooling or average-pooling to directly predict the score of an image will lose crucial information.\nTherefore, we use an adaptive weighted strategy to predict the score of an image.\n\n\nIn this work, we introduce an effective hybrid architecture for image quality assessment, which leverages local details from a shallow CNN and global semantic information captured by ViT to further improve IQA accuracy. Specifically, we first adopt a two-branch feature extractor. Then, we use semantic information captured by ViT to find the salient region in images through deformable convolution~\\cite{dai2017deformable}. Based on the consideration that each pixel in the deep feature map corresponds to different patches of the input image, we introduce the patch-wise prediction module, which contains two branches, one to calculate a score for each image patch, the other one to calculate the weight of each score.\n\nExtensive experiments show that our method outperforms current approaches in four benchmark image quality assessment datasets~\\cite{sheikh2006statistical,larson2010most,ponomarenko2015image,jinjin2020pipal}. The scatter diagram of the correlation between predicted scores and MOS is shown in \\cref{fig:scatter} where the plot for IQT is from our own implementation. Visualization experiments reveal that the proposed method is almost linear with MOS, which means that we can better imitate human image perception. Our primary contributions can be summarized as follows:\n\\begin{itemize}\n\\item We propose an effective hybrid architecture for image quality assessment, which compares images at the patch level, adds spatial details as a supplement, and scores images patch by patch, considering the relationship between patches and different contributions from each patch.\n\n\\item Our method outperforms the state-of-the-art approaches on four benchmark image quality assessment datasets. In particular, the proposed architecture achieves outstanding performance on the PIPAL dataset with various GAN-based distortion and ranked first in the NTIRE 2022 challenge on perceptual image quality assessment.\n\\end{itemize}\n\n\\section{Related Works}\n\\label{sec:related}\n\\subsection{Image Quality Assessment}\nThe goal of IQA is to mimic the HVS to rate the perceived quality of an image accurately. Although it's easy for human beings to assess an image's perceptual quality, IQA is considered to be difficult for machines. Depending on the scenarios and conditions, current IQA methods can be divided into three categories: full-reference (FR) ,and no-reference (NR) IQA. FR-IQA methods take the distortion image and the corresponding reference image as inputs to measure their perceptual similarity. The most widely-used FR-IQA metrics are PSNR and SSIM~\\cite{wang2004image} which are conventional and easy to optimize.\n\\begin{figure*}[th]\n \\centering\n \\includegraphics[scale=0.48]{architecture_final.pdf}\n \\caption{Overview of AHIQ. The proposed model takes a pair of the reference image and distortion image as input and then obtains feature maps through ViT~\\cite{dosovitskiy2020image} and CNN, respectively. The feature maps of reference image from ViT are used as global information to obtain the offset map of the deformable convolution\\cite{dai2017deformable}. After the feature fusion module which fuses the feature maps, we use a patch-wise prediction module to predict a score for each image patch. The final output is the weighted sum of the scores.}\n \\label{fig1:arch}\n\\end{figure*}\nApart from the conventional IQA methods, various learning-based FR-IQA methods~\\cite{zhang2018unreasonable,bosse2017deep,prashnani2018pieapp} have been proposed to address the limitations of conventional IQA methods recently. Zhang \\etal~\\cite{zhang2018unreasonable} proposed to use the learned perceptual image patch similarity (LPIPS) metric for FR-IQA and proved that deep features obtained through pre-trained DNNs outperform previous classic metrics by large margins. WaDIQaM~\\cite{bosse2017deep} is a general end-to-end deep neural network that enables jointly learning of local quality and local weights. PieAPP~\\cite{prashnani2018pieapp} is proposed to learn to rank rather than learn to score, which means the network learns the probability of preference of one image over another. IQT~\\cite{cheon2021perceptual} applies an encoder-decoder transformer architecture with trainable extra quality embedding and ranked first place in NTIRE 2021 perceptual image quality assessment challenge. In addition, common CNN-based NR-IQA methods~\\cite{su2020blindly,wu2020end,xia2020domain} directly extract features from the low-quality images and outperform traditional handcrafted approaches. You \\etal~\\cite{ you2021transformer} introduced transformer architecture for the NR-IQA recently.\n\n\n\n\n\\subsection{Vision Transformer}\nTransformer architecture based on self-attention mechanism~\\cite{vaswani2017attention} was first proposed in the field of Natural Language Processing (NLP) and significantly improved the performances of many NLP tasks thanks to its representation capability. Inspired by its success in NLP, efforts are made to apply transformers to vision tasks such as image classification~\\cite{dosovitskiy2020image}, object detection~\\cite{carion2020end, zhu2020deformable}, low-level vision~\\cite{yang2020learning}, \\etc. Vision Transformer (ViT) introduced by Dosovitskiy~\\etal~\\cite{dosovitskiy2020image} is directly inherited from NLP, but takes raw image patches as input instead of word sequences. ViT and its follow-up studies have become one of the mainstream feature extraction backbones except for CNNs.\n\nCompared with the most commonly used CNNs, transformer can derive global information while CNNs mainly focus on local features. In IQA tasks, global and local information are both crucial to the performance because when human beings assess image quality, both the information are naturally taken into account. Inspired by this assumption, we propose to combine long-distance features and local features captured by ViT and CNNs, respectively. To fulfill this goal, we use a two-branch feature extraction backbone and feature fusion modules, which will be detailed in~\\cref{sec:method}. \n\n\n\\subsection{Deformable Convolution}\nDeformable convolution~\\cite{dai2017deformable} is an efficient and powerful mechanism which is first proposed to deal with sparse spatial locations in high-level vision tasks such as object detection~\\cite{dai2017deformable,bertasius2018object, zhu2019deformable }, semantic segmentation~\\cite{zhu2019deformable}, and human pose estimation~\\cite{sun2018integral}. By using deformed sampling locations with learnable offsets, deformable convolution enhances the spatial sampling locations and improves the transformation modeling ability of CNNs. Recently, deformable convolution continues its strong performance in low-level vision tasks including video deblurring~\\cite{wang2019edvr}, video super-resolution~\\cite{chan2021understanding}. It is first combined with IQA methods by Shi~\\etal~\\cite{shi2021region} to perform a reference-oriented deformable convolution in the full-reference scenario.\n\n\n\n\\section{Methodology}\n\\label{sec:method}\n\nIn this section, we introduce the overall framework of the Attention-based Hybrid Image Quality Assessment Network (AHIQ). As shown in Fig \\ref{fig1:arch}, the proposed network takes pairs of reference images and distortion images as input, and it consists of three key components: a feature extraction module, a feature fusion module, and a patch-wise prediction module. \n\nFor the reason that GAN-based image restoration methods~\\cite{wang2018esrgan,gu2020image} often fabricate plausible details and textures, it is difficult for the network to distinguish GAN-generated texture from noise and real texture by pixel-wise image difference. Our proposed model aims to deal with it. We employ the Vision Transformer to model the relationship and capture long-range dependencies among patches. Shallow CNN features are introduced to add detailed spatial information. In order to help CNN focus on the salient region, we use deformable convolution guided by semantic information from ViT. We use an adaptive weighted scoring mechanism to give a comprehensive assessment.\n\n\n\n\\subsection{Feature Extraction Module}\n\\label{subsec:extraction}\n\\begin{figure}[th]\n\\centering\n\\includegraphics[scale=0.41]{architecture_feat.pdf}\n\\caption{The illustration of vision Transformer for feature extraction module. The class token (orange) is regarded when the feature maps are extracted.\n}\n\\label{fig2:feat}\n\\end{figure}\nAs is depicted in \\cref{fig1:arch}, the front part of the architecture is a two-branch feature extraction module that consists of a ViT branch and a CNN branch. The transformer feature extractor mainly focuses on extracting global and semantic representations. Self-attention modules in transformer enable the network to model long-distance features and encode the input image patches into feature representations. Patch-wise encoding is helpful to assess the output image quality of GAN-based image restoration because it enhances the tolerance of spatial misalignment. Since humans also pay attention to details when judging the quality of an image, so detailed and local information is also important. To this end, we introduce another CNN extraction branch apart from the transformer branch to add more local textures.\n\nIn the forward process, a pair of the reference image and distortion image are fed into the two branches, respectively, and we then take out their feature maps in the early stages. For the transformer branch, as illustrated in~\\cref{fig2:feat}, output sequences from Vision Transformer~\\cite{dosovitskiy2020image} are reshaped into feature maps $f_{T}\\in \\mathbb{R}^{p\\times p \\times 5c}$ discarding the class token, where $p$ represent the size of the feature map. For the CNN branch, we extract shallow feature map from ResNet~\\cite{he2016deep} $f_{C}\\in \\mathbb{R}^{4p\\times 4p \\times C}$ where $C=256\\times 3$. Finally, we put the obtained feature maps into the feature fusion module, which will be specified next.\n \n\n\\subsection{Feature Fusion Module}\n\\label{subsec:fusion}\nWe argue that feature maps from the early stages of CNN provide low-level texture details but bring along some noise. To address this problem, we take advantage of transformer architecture to capture global and semantic information. In our proposed network, feature maps from ViT with rich semantic information are used to find the salient region of the image. This perception procedure is performed in a content-aware manner and allows the network better mimic the way humans perceive image quality. Particularly, the feature maps from ViT are used to learn an offset map for deformable convolution as is shown in~\\cref{fig2:feat}. Then we perform this deformable convolution~\\cite{dai2017deformable} operation on feature maps from CNN, which we elaborate on previously. In this way, features from a shallow CNN can be better modified and utilized for further feature fusion. Obviously, in the previous description, feature maps from the two branches differ from each other in spatial dimension and need to be aligned. Therefore, a simple 2-layer convolution network is applied to project the feature maps after deformable convolution to the same width~\\textit{W} and height~\\textit{H} with ViT. The whole process can be formulated as follows:\n\\begin{align}\n && \\Delta p &= \\text{Conv1}(f_{T})), \\\\\n && f_{C} &= \\text{DConv}(f_{org},\\Delta p), \\\\\n && f_{C}^{'} &= \\text{Conv2}(\\text{ReLU}(\\text{Conv2}(f_{C}))), \\\\\n && f^{u} &= \\text{Concat}[f_{T},f_{C}^{'}], \\\\\n && f_{all} &= \\text{Concat}[f^{u}_{dis},f^{u}_{ref},f^{u}_{dis}-f^{u}_{ref}], \\\\\n && f_{out} &= \\text{Conv3}(\\text{ReLU}(\\text{Conv3}(f_{all}))),\n\\end{align}\nwhere $f_T$ denotes feature maps from the transformer branch, $\\Delta p$ denotes offset map, $f_{org}$ and $f_C$ denote feature maps from CNN, DConv means deformable convolution. Note that Conv2 is a convolution operation with a stride of 2, downsampling $f_C \\in \\mathbb{R}^{4p \\times 4p \\times C}$ by four times to $f_C^{'} \\in \\mathbb{R}^{p \\times p \\times C}$. \n\n\\subsection{Patch-wise Prediction Module}\n\\label{subsec:pooling}\nGiven that each pixel in the deep feature map corresponds to a different patch of the input image and contains abundant information, the information in the spatial dimension is indispensable. However, in previous works, spatial pooling methods such as max-pooling and average-pooling are applied to obtain a final single quality score. This pooling strategy loses some information and ignores the relationships between image patches. Therefore, we introduce a two-branch patch-wise prediction module which is made up of a prediction branch and a spatial attention branch, as illustrated in \\cref{fig:pixel}. The prediction branch calculates a score for each pixel in the feature map, while the spatial attention branch calculates an attention map for each corresponding score. Finally, we can obtain the final score by weighted summation of scores. The weighted sum operation helps to model the significance of the region to simulate the human visual system. This can be expressed as follows:\n\\begin{equation}\n s_f = \\frac{\\textbf{s} * \\textbf{w}}{\\sum \\textbf{w}},\n\\end{equation}\nwhere $\\textbf{s}\\in \\mathbb{R}^{H\\times W \\times 1}$ denotes score map, $\\textbf{w}\\in \\mathbb{R}^{H\\times W \\times 1}$ denotes the corresponding attention map, $*$ means Hadamard product and $s_f$ means the final predicted score. MSE loss between the predicted score and the ground truth score is utilized for the training process in our proposed method.\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[scale=0.43]{architecture_patch.pdf}\n\\caption{The pipeline of the proposed patch-wise prediction module. This two-branch module takes feature maps as input, then generates a patch-wise score map and its corresponding attention map to obtain the final prediction by weighted average.}\n\\label{fig:pixel}\n\\end{figure}\n\n\\begin{table*}[th]\n\\centering\n\\renewcommand{\\arraystretch}{1.08}\n\\caption{IQA datasets for performance evaluation and model training.}\n\\begin{tabular}{cccccccc}\n\\toprule[1.2pt]\nDatabase & \\# Ref & \\# Dist & Dist. Type & \\# Dist. Type & Rating & Rating Type & Env. \\\\ \\hline\nLIVE~\\cite{sheikh2006statistical} & 29 & 779 & traditional & 5 & 25k & MOS & lab \\\\\nCSIQ~\\cite{larson2010most} & 30 & 866 & traditional & 6 & 5k & MOS & lab \\\\\nTID2013~\\cite{ponomarenko2015image} & 25 & 3,000 & traditional & 25 & 524k & MOS & lab \\\\\nKADID-10k~\\cite{lin2019kadid} & 81 & 10.1k & traditional & 25 & 30.4k & MOS & crowdsourcing \\\\\nPIPAL~\\cite{jinjin2020pipal} & 250 & 29k & trad.+alg.outputs & 40 & 1.13m & MOS & crowdsourcing \\\\ \\toprule[1.2pt]\n\\label{tab:data}\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[th]\n\\centering\n\\caption{Performance comparisons on LIVE, CSIQ, and TID2013 Databases. Performance scores of other methods are as reported in the corresponding original papers and~\\cite{ding2020image}. The best scores are~\\textbf{bolded} and missing scores are shown as ``\u2013'' dash.}\n\\setlength{\\tabcolsep}{6mm}{\n\\begin{tabular}{ccccccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c}{LIVE} & \\multicolumn{2}{c}{CSIQ} & \\multicolumn{2}{c}{TID2013} \\\\ \\cline{2-7} \n & PLCC & SROCC & PLCC & SROCC & PLCC & SROCC \\\\ \\hline\nPSNR & 0.865 & 0.873 & 0.819 & 0.810 & 0.677 & 0.687 \\\\\nSSIM~\\cite{wang2004image} & 0.937 & 0.948 & 0.852 & 0.865 & 0.777 & 0.727 \\\\\nMS-SSIM~\\cite{wang2003multiscale} & 0.940 & 0.951 & 0.889 & 0.906 & 0.830 & 0.786 \\\\\nFSIMc~\\cite{zhang2011fsim} & 0.961 & 0.965 & 0.919 & 0.931 & 0.877 & 0.851 \\\\\nVSI~\\cite{zhang2014vsi} & 0.948 & 0.952 & 0.928 & 0.942 & 0.900 & 0.897 \\\\\nMAD~\\cite{larson2010most} &0.968 &0.967 &0.950 &0.947 &0.827 &0.781 \\\\\nVIF~\\cite{sheikh2006image} &0.960 &0.964 &0.913 &0.911 &0.771 &0.677 \\\\\nNLPD~\\cite{laparra2016perceptual} &0.932 &0.937 &0.923 &0.932 &0.839 &0.800 \\\\\nGMSD~\\cite{xue2013gradient} &0.957 &0.960 &0.945 &0.950 &0.855 &0.804\\\\ \nSCQI~\\cite{bae2016novel} & 0.937 & 0.948 & 0.927 & 0.943 & 0.907 & 0.905 \\\\ \\hline\nDOG-SSIMc~\\cite{pei2015image} & 0.966 & 0.963 & 0.943 & 0.954 & 0.934 & 0.926 \\\\\nDeepQA~\\cite{kim2017deep} & 0.982 & 0.981 & 0.965 & 0.961 & 0.947 & 0.939 \\\\\nDualCNN~\\cite{varga2020composition} & - & - & - & - & 0.924 & 0.926 \\\\\nWaDIQaM-FR~\\cite{bosse2017deep} & 0.98 & 0.97 & - & - & 0.946 & 0.94 \\\\\nPieAPP~\\cite{prashnani2018pieapp} & 0.986 & 0.977 & 0.975 & 0.973 & 0.946 & 0.945 \\\\\nJND-SalCAR~\\cite{seo2020novel} & 0.987 & \\textbf{0.984} & 0.977 & \\textbf{0.976} & 0.956 & 0.949 \\\\\nAHIQ (ours) & \\textbf{0.989} & \\textbf{0.984} & \\textbf{0.978} & 0.975 & \\textbf{0.968} & \\textbf{0.962} \\\\ \\toprule[1.2pt]\n\n\\end{tabular}}\n\\label{tab:sota}\n\\end{table*}\n\n\\section{Experiment}\n\\subsection{Datasets}\n\nWe employ four datasets that are commonly used in the research of perceptual image quality assessment, including LIVE~\\cite{sheikh2006statistical}, CSIQ~\\cite{larson2010most}, TID2013~\\cite{ponomarenko2015image}, and PIPAL~\\cite{jinjin2020pipal}. \\cref{tab:data} compares the listed datasets in more detail. In addition to PIPAL, the other datasets only include traditional distortion types, while PIPAL includes a large number of distorted images including GAN-generated images. \n\nAs recommended, we randomly split the datasets into training (60\\%), validation (20\\%), and test set (20\\%) according to reference images. Therefore, the test data and validation data will not be seen during the training procedure. We use the validation set to select the model with the best performance and use the test set to evaluate the final performance.\n\n\\subsection{Implementation Details}\nSince we use ViT~\\cite{dosovitskiy2020image} and ResNet~\\cite{he2016deep} models pre-trained on ImageNet~\\cite{ILSVRC15}, we normalize all input images and randomly crop them into $224\\times224$. We use the outputs of five intermediate blocks $\\{0,1,2,3,4\\}$ in ViT, each of which consists of a self-attention module and a Feed-Forward Network (FFN). The feature map from one block $f\\in \\mathbb{R}^{p\\times p \\times c}$, where $c=768, p=14 \\ \\text{or} \\ 28$, are concatenated into $f_{T}\\in \\mathbb{R}^{p\\times p \\times 6c}$. We also take out the output feature maps from all the 3 layers in stage 1 of ResNet and concatenate them together to get $f_{C}\\in \\mathbb{R}^{56\\times 56 \\times C}$ where $C=256\\times 3$. And random horizontal flip rotation is applied during the training. The training loss is computed using a mean squared error (MSE) loss function. During the validation phase and test phase, we randomly crop each image 20 times and the final score is the average score of each cropped image. It should be noted that we use pretrained ViT-B\/16 as the backbone in all experiments on traditional datasets including LIVE, CSIQ and TID2013, while ViT-B\/8 is utilized in PIPAL.\n\nFor optimization, we use the AdamW optimizer with an initial learning rate $lr$ of $10^{-4}$ and weight decay of $10^{-5}$. We set the minibatch size as 8. Set the learning rate of each parameter group using a cosine annealing schedule, where $\\eta_{max}$ is set to the initial $lr$ and the number of epochs $T_{cur}$ is 50. \nWe implemented our proposed model AHIQ in Pytorch and trained using a single NVIDIA GeForce RTX2080 Ti GPU. The practical training runtimes differ across datasets as the number of images in each dataset is different. Training one epoch on the PIPAL dataset requires thirty minutes.\n\n\\subsection{Comparison with the State-of-the-art Methods}\nWe assess the performance of our model with Pearson's linear\ncorrelation coefficient (PLCC) and Spearman's rank-order correlation coefficient (SROCC). \nPLCC assesses the linear correlation between ground truth and the predicted quality scores, whereas SROCC describes the level of monotonic correlation.\n\n\\vspace{2pt}\n\\noindent\\textbf{Evaluation on Traditional Dataset.} We evaluate the effectiveness of AHIQ on four benchmark datasets. For all our tests, we follow the above experimental setup. It can be shown in~\\cref{tab:sota} that AHIQ outperforms or is competitive with WaDIQaM~\\cite{bosse2017deep}, PieAPP~\\cite{prashnani2018pieapp}, and JND-SalCAR~\\cite{seo2020novel} for all tested datasets. Especially on the more complex dataset TID2013, our proposed model achieved a solid improvement over previous work. This shows that the AHIQ can cope well with different types of distorted images.\n\\begin{table}[th]\n\\centering\n\\caption{Performance comparison after training on the entire KADID dataset~\\cite{lin2019kadid}, then test on LIVE, CSIQ, and TID2013 Databases. Part of the performance scores of other methods are borrowed from~\\cite{ding2020image}. The best scores are~\\textbf{bolded} and missing scores are shown as ``\u2013'' dash.}\n\\scalebox{0.82}{\n\\begin{tabular}{cccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{Method} & LIVE & CSIQ & TID2013 \\\\ \\cline{2-4} \n& PLCC\/SROCC & PLCC\/SROCC & PLCC\/SROCC \\\\ \\hline\n\nWaDIQaM~\\cite{bosse2017deep} & 0.940\/0.947 & 0.901\/0.909 & 0.834\/0.831 \\\\\nPieAPP~\\cite{prashnani2018pieapp} & 0.908\/0.919 & 0.877\/0.892 & 0.859\/0.876 \\\\\nLPIPS~\\cite{zhang2018unreasonable} & 0.934\/0.932 &0.896\/0.876 &0.749\/0.670 \\\\\nDISTS~\\cite{ding2020image} & \\textbf{0.954}\/0.954 & 0.928\/0.929 & 0.855\/0.830 \\\\\nIQT~\\cite{cheon2021perceptual} & -\/\\textbf{0.970} & -\/0.943 & -\/0.899 \\\\\nAHIQ (ours) & 0.952\/\\textbf{0.970} & \\textbf{0.955\/0.951} & \\textbf{0.899\/0.901} \\\\ \\toprule[1.2pt]\n\\end{tabular}\n}\n\\label{tab:kadid}\n\\end{table}\n\n\\vspace{2pt}\n\\noindent\\textbf{Evaluation on PIPAL.} We compare our models with the state-of-the-art FR-IQA methods on the NTIRE 2022 IQA challenge validation and testing datasets. As shown in \\cref{tab:pipal}, AHIQ achieves outstanding performance in terms of PLCC and SROCC compared with all previous work. In particular, our method substantially outperforms IQT, which is recognized as the first transformer-based image quality assessment network, through the effective feature fusion from the shallow CNN and ViT as well as the proposed patch-wise prediction module. This verifies the effectiveness of our model for GAN-based distortion image quality assessment.\n\n\\begin{table}[th]\n\\centering\n\\caption{\nPerformance comparison of different IQA methods on PIPAL dataset. AHIQ-C is the ensemble version we used for the NTIRE 2022 Perceptual IQA Challenge.\n}\n\\begin{tabular}{ccccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c}{Validation} & \\multicolumn{2}{c}{Test} \\\\ \\cline{2-5} \n & PLCC & SROCC & PLCC & SROCC \\\\ \\hline\nPSNR & 0.269 & 0.234 & 0.277 & 0.249 \\\\\nNQM~\\cite{damera2000image} & 0.364 & 0.302 & 0.395 & 0.364 \\\\\nUQI~\\cite{wang2002universal} & 0.505 & 0.461 & 0.450 & 0.420 \\\\\nSSIM~\\cite{wang2004image} & 0.377 & 0.319 & 0.391 & 0.361 \\\\\nMS-SSIM~\\cite{wang2003multiscale} & 0.119 & 0.338 & 0.163 & 0.369 \\\\\nRFSIM~\\cite{zhang2010rfsim} & 0.285 & 0.254 & 0.328 & 0.304 \\\\\nGSM~\\cite{liu2011image} & 0.450 & 0.379 & 0.465 & 0.409 \\\\\nSRSIM~\\cite{zhang2012sr} & 0.626 & 0.529 & 0.636 & 0.573 \\\\\nFSIM~\\cite{zhang2011fsim} & 0.553 & 0.452 & 0.571 & 0.504 \\\\\nVSI~\\cite{zhang2014vsi} & 0.493 & 0.411 & 0.517 & 0.458 \\\\\nNIQE~\\cite{mittal2012making} & 0.129 & 0.012 & 0.132 & 0.034 \\\\\nMA~\\cite{ma2017learning} & 0.097 & 0.099 & 0.147 & 0.140 \\\\\nPI~\\cite{blau2018perception} & 0.134 & 0.064 & 0.145 & 0.104 \\\\\nBrisque~\\cite{mittal2011blind} & 0.052 & 0.008 & 0.069 & 0.071 \\\\ \\hline\nLPIPS-Alex~\\cite{zhang2018unreasonable} & 0.606 & 0.569 & 0.571 & 0.566 \\\\\nLPIPS-VGG~\\cite{zhang2018unreasonable} & 0.611 & 0.551 & 0.633 & 0.595 \\\\\nDISTS~\\cite{ding2020image} & 0.634 & 0.608 & 0.687 & 0.655 \\\\\nIQT~\\cite{cheon2021perceptual} & 0.840 & 0.820 & 0.799 & 0.790 \\\\ \\hline\nAHIQ (ours) & 0.845 & 0.835 & 0.823 & 0.813 \\\\\nAHIQ-C (ours) & \\textbf{0.865} & \\textbf{0.852} & \\textbf{0.828} & \\textbf{0.822} \\\\ \\toprule[1.2pt]\n\\end{tabular}\n\\label{tab:pipal}\n\\end{table}\n\n\\vspace{2pt}\n\\noindent\\textbf{Cross-Database Performance Evaluation.} \nTo evaluate the generalization of our proposed AHIQ, we conduct the cross-dataset evaluation on LIVE, CSIQ, and TID2013. We train the model on KADID and the training set of PIPAL respectively. Then we test it on the full set of the other three benchmark datasets. As shown in \\cref{tab:kadid} and \\cref{tab:cross}, AHIQ achieves satisfactory generalization ability. \n\n\\begin{table}[th]\n\\centering\n\\caption{Performance comparison for cross-database evaluations.}\n\\scalebox{0.82}{\n\\begin{tabular}{cccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{Method} & LIVE & CSIQ & TID2013 \\\\ \\cline{2-4} \n & PLCC\/SROCC & PLCC\/SROCC & PLCC\/SROCC \\\\ \\hline\nPSNR & 0.865\/0.873 & 0.786\/0.809 & 0.677\/0.687 \\\\\nWaDIQaM~\\cite{bosse2017deep} & 0.837\/0.883 & -\/- & 0.741\/0.698 \\\\\nRADN~\\cite{shi2021region} & 0.878\/0.905 & -\/- & 0.796\/0.747 \\\\\nAHIQ (ours) & \\textbf{0.911\/0.920} & \\textbf{0.861\/0.865} & \\textbf{0.804\/0.763} \\\\ \\toprule[1.2pt]\n\\end{tabular}\n}\n\\label{tab:cross}\n\\end{table}\n\n\n\n\\subsection{Ablation Study}\nIn this section, we analyze the effectiveness of the proposed network by conducting ablation studies on the NTIRE 2022 IQA Challenge testing datasets~\\cite{gu2022ntire}. With different configuration and implementation strategies, we evaluate the effect of each of the three major components: feature extraction module, feature fusion module, and patch-wise prediction module.\n\\begin{table}[ht]\n\\centering\n\\caption{Comparison of different feature fusion strategies on the NTIRE 2022 IQA Challenge testing datasets. CNN refers to Resnet50 and ViT refers to ViT-B\/8 in this experiment. }\n\n\\begin{tabular}{cccccc}\n\\toprule[1.2pt]\n\\multirow{2}{*}{No.} & \\multicolumn{2}{c}{Feature} & \\multirow{2}{*}{Fusion Method} & \\multirow{2}{*}{PLCC} & \\multirow{2}{*}{SROCC} \\\\ \\cline{2-3}\n & CNN & ViT & & & \\\\ \\hline\n1 & \\checkmark & \\checkmark & deform+concat & \\textbf{0.823} & \\textbf{0.813} \\\\\n2 & \\checkmark & \\checkmark & concat & 0.810 & 0.799 \\\\\n3 & \\checkmark & & - & 0.792 & 0.789 \\\\\n4 & & \\checkmark & - & 0.799 & 0.788 \\\\ \\toprule[1.2pt]\n\\end{tabular}\n\\label{tab:fusion}\n\\end{table}\n\n\n\\vspace{2pt}\n\\noindent\\textbf{Feature Extraction Backbone.} We experiment with different representative feature-extraction backbones and the comparison result is provided in \\cref{tab:backbone}. The CNN backbones used for comparison include ResNet50, ResNet101, ResNet152~\\cite{he2016deep}, HRNet~\\cite{wang2020deep}, Inception-ResNet-V2~\\cite{szegedy2017inception}, and the transformer backbones include ViT-B\/16 and ViT-B\/8~\\cite{dosovitskiy2020image}. It is noteworthy that ViT-B consists of 12 transformer blocks and the sizes of the image patches are $16\\times 16$ and $8\\times 8$ respectively with an input shape of $224\\times 224$. \n\nIt can be found that the network using ResNet50 and ViT-B\/8 ends up performing the best. The experimental results demonstrate that deeper and wider CNN is unnecessary for AHIQ. We believe this is because CNN plays the role of providing shallow and local feature information in AHIQ. We only take out the intermediate layers from the first stage, so shallow features will contain less information when the network is too deep or too complicated.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Comparison of different feature extraction backbones on the NTIRE 2022 IQA Challenge testing datasets.}\n\\scalebox{0.95}{\n\\begin{tabular}{ccccc}\n\\toprule[1.2pt]\nCNN & ViT & PLCC & SROCC & Main Score \\\\ \\hline\nResnet50 & \\multirow{5}{*}{ViT-B\/8} & \\textbf{0.823} & \\textbf{0.813} & \\textbf{1.636} \\\\\nResnet101 & & 0.802 & 0.788 & 1.590 \\\\\nResnet152 & & 0.807 & 0.793 & 1.600 \\\\\nHRnet & & 0.806 & 0.796 & 1.601 \\\\\nIncepResV2 & & 0.806 & 0.793 & 1.599 \\\\ \\hline\nResnet50 & ViT-B\/16 & 0.811 & 0.803 & 1.614 \\\\ \\toprule[1.2pt]\n\\end{tabular}}\n\\label{tab:backbone}\n\\end{table}\n\n\\vspace{2pt}\n\\noindent\\textbf{Fusion strategy.} We further examine the effect of features from CNN and ViT as well as the feature fusion strategies. As is tabulated in \\cref{tab:fusion}, the first two experiments adopt different methods for feature fusion. The first one is the method we adopt in our AHIQ. For the second experiment, the features from transformer and ViT are simply concatenated together. The first method outperforms the second one by a large margin which demonstrates that using deformable convolution to modify CNN feature maps is well-effective. This further illustrates the power of global and semantic information in transformer to guide the shallow features by paying more attention to the salient regions.\n\nWe also conduct ablation studies on using features from ViT and from CNN separately. Results are at the last two rows in~\\cref{tab:fusion}. One can observe that only using one of the CNN and Transformer branches results in a dramatic decrease in performance. This experimental result shows that both global semantic information brought by ViT and local texture information introduced by CNN is very crucial in this task, which is well consistent with our previous claim.\n\\begin{figure}[th]\n\\centering\n\\includegraphics[scale=0.26]{optical_flow2.pdf}\n\\caption{The visualization of learned offsets from deformable convolution. For each case, the vector flow which displays the learned offsets and zoomed-in details are included.}\n\\label{fig:optical}\n\\end{figure}\n\n\\vspace{2pt}\n\\noindent\\textbf{Visualization of Learned Offset.} We visualize the learned offsets from deformable convolution in \\cref{fig:optical}. It can be observed that the learned offsets indicated by arrows mainly affect edges and salient regions. In addition, most of the offset vectors point from the background to the salient regions, which means that in the process of convolution, the sampling locations moves to the significant region by the learned offsets. This visualization results illustrate the argument we made earlier that semantic information from ViT help CNN see better by deformable convolution.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Comparison of different pooling strategy on the NTIRE 2022 IQA Challenge testing datasets. Note that ``Patch'' denotes the patch-wise prediction and ``Spatial'' denotes the spatial pooling.}\n\\begin{tabular}{cccc}\n\\toprule[1.2pt]\nPooling Strategy & PLCC & SROCC & Main Score \\\\ \\hline\nPatch & \\textbf{0.823} & \\textbf{0.813} & \\textbf{1.636} \\\\\nSpatial & 0.794 & 0.795 & 1.589 \\\\\nPatch + Spatial & 0.801 & 0.791 & 1.593\\\\ \\toprule[1.2pt]\n\\end{tabular}\n\\label{tab:pooling}\n\\end{table}\n\n\n\\vspace{2pt}\n\\noindent\\textbf{Pooling Strategy.} Experiments on different pooling strategies are conducted, and the results are shown in \\cref{tab:pooling}. We first perform patch-wise prediction, which is elaborated in \\cref{subsec:pooling}. For comparison, we follow WaDIQaM~\\cite{bosse2017deep} and IQMA~\\cite{guo2021iqma} to use spatial pooling that combines max-pooling and average-pooling in spatial dimension to obtain a score vector $S\\in \\mathbb{R}^{1\\times 1 \\times C}$. The final score is the weighted sum of the score vector and the final result is shown in the second row of \\cref{subsec:pooling}. Then we try to combine the previous two pooling method and propose to use the average of the output score from patch-wise prediction and spatial pooling in the third experiment. Patch-wise prediction module proposed in AHIQ performs better than the other two, and experimental results further prove the validity of the patch-wise prediction operation. It confirms our previous claim that different regions should contribute differently to the final score.\n\n\n\\subsection{NTIRE 2022 Perceptual IQA Challenge}\nThis work is proposed to participate in the NTIRE 2022 perceptual image quality assessment challenge~\\cite{gu2022ntire}, the objective of which is to propose an algorithm to estimate image quality consistent with human perception. The final results of the challenge in the testing phase are shown in~\\cref{tab:final}. Our ensemble approach won the first place in terms of PLCC, SROCC, and main score.\n\\begin{table}[ht]\n\\centering\n\\caption{The results of NTIRE 2022 challenge FR-IQA track on the testing dataset. This table only shows part of the participants and best scores are~\\textbf{bolded}.}\n\\begin{tabular}{cccc}\n\\toprule[1.2pt]\nMethod & PLCC & SROCC & Main Score \\\\ \\hline\nOurs\n& \\textbf{0.828} & \\textbf{0.822} & \\textbf{1.651} \\\\\n\\nth{2} \n& 0.827 & 0.815 & 1.642 \\\\\n\\nth{3} & 0.823 & 0.817 & 1.64 \\\\\n\\nth{4} & 0.775 & 0.766 & 1.541 \\\\\n\\nth{5} & 0.772 & 0.765 & 1.538 \\\\ \\toprule[1.2pt]\n\\end{tabular}\n\\label{tab:final}\n\\end{table}\n\n\n\\section{Conclusion}\nIn this paper, we propose a novel network called Attention-based Hybrid Image Quality Assessment Network (AHIQ), for the full-reference image quality assessment task. The proposed hybrid architecture takes advantage of the global semantic features captured by ViT and local detailed textures from a shallow CNN during feature extraction. To help CNN pay more attention to the salient region in the image, semantic information from ViT is adopted to guide deformable convolution so that model can better mimic how humans perceive image quality. Then we further propose a feature fusion module to combine different features. We also introduce a patch-wise prediction module to replace spatial pooling and preserve information in the spatial dimension. Experiments show that the proposed method not only outperforms the state-of-the-art methods on standard datasets, but also has a strong generalization ability on unseen samples and hard samples, especially GAN-based distortions. The ensembled version of our method ranked first place in the FR track of the NTIRE 2022 Perceptual Image Quality Assessment Challenge.\n\n\\vspace{2pt}\n\\noindent\\textbf{Acknowledgment.} This work was supported by the Key Program of the National Natural Science Foundation of China under Grant No. U1903213 and the Shenzhen Key Laboratory of Marine IntelliSense and Computation under Contract ZDSYS20200811142605016.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nGraphene, a single atomic layer of carbon arranged in a honeycomb lattice, holds great promise for numerous applications due to its remarkable mechanical, optical, and electronic properties and serves as a powerful material platform for studying relativistic Dirac fermions due to its linearly dispersing bands\\cite{novoselov_two-dimensional_2005,geim_rise_2007, castro_neto_electronic_2009}. Graphene was also the first material in which a member of the striking class of macroscopic quantum phenomena\\cite{anderson_observation_1995,osheroff_evidence_1972,willett_observation_1987,bardeen_theory_1957} -- the quantum Hall effect (QHE)\\cite{klitzing_new_1980} -- could be observed at room temperature, when subject to large magnetic fields\\cite{novoselov_room-temperature_2007}. In the quantum Hall state, charge carriers are forced into cyclotron orbits with quantized radii and energies known as Landau levels (LLs), once subjected to the influence of a magnetic field. In order to observe this effect, certain conditions must be met: The magnetic field must be large enough that the resulting spacing between LLs is larger than the thermal energy $(\\Delta E_{LL}>k_BT)$; the charge carrier lifetime between scattering events must be longer than the characteristic time of the cyclotron orbit $(t_\\textrm{life}>1\/\\omega _c)$; and the magnetic field must be uniform on length scales greater than the LL orbit. This typically mandates the need for cryogenic temperatures, clean materials, and large applied magnetic fields. Dirac fermions in graphene provide a way to lift these restrictions: Under certain strain patterns, graphene's electrons behave as if they were under the influence of large magnetic fields, without applying an actual field from outside the material\\cite{guinea_energy_2010,levy_strain-induced_2010,liu_tailoring_2018,rechtsman_strain-induced_2013}. These so-called pseudomagnetic fields only couple to the relativistic electrons around the Dirac point and, under the QHE conditions above, lead to the formation of flat, quantized LLs. This has been successfully observed using a range of methods\\cite{levy_strain-induced_2010,liu_tailoring_2018,rechtsman_strain-induced_2013}, but was so far restricted to small regions, which severely limits its applicability.\n\n\\section*{Results}\n\n\\begin{figure}\n\\makebox[\\textwidth]{\\includegraphics[width=183mm]{F2_AFM_v8.png}}\n\\caption{\\textbf{Identification of strained nanoprisms.} (\\textbf{A}) Horizontal derivative AFM topography image of our monolayer graphene grown on a SiC substrate. Triangular nanoprisms are dispersed on the surface. Inset: AFM topography image of the same area. Substrate terrace steps are about $10\\ \\textrm{nm}$ in height. (\\textbf{B}) Top: Close-up AFM topography of the area indicated by the black box in (\\textbf{A}). Bottom: Line cut through the AFM data marked by the purple line in the close-up. (\\textbf{C}) Overview STM topography image (200 nm x 200 nm, $V_{sample}=100\\ \\textrm{mV}$, $I_{tun.}=2\\ \\textrm{pA}$) showing a single nanoprism. (\\textbf{D}) Schematic structure of 6H-SiC, showing its layered ABCACB stacking order with epitaxial graphene on top (yellow). Inside the nanoprism a single layer \\emph{within} the unit cell is missing, exposing the graphene to a different substrate surface termination, as illustrated in the top view. The carbon buffer layer is not shown for clarity. (\\textbf{E}) Atomically resolved STM images (10\\,nm x 10\\,nm, $V_{sample}=30\\ \\textrm{mV}$, $I_{tun.}=2\\ \\textrm{pA}$) inside (top) and outside (bottom) of the nanoprism. (\\textbf{F}) Difference map of the two Fourier transformed (FT) images in (\\textbf{E}) visualizing the strain pattern inside the nanoprism.}\n\\label{fig:f2_afm}\n\\end{figure}\n\nHere, we directly visualize the formation of flat LLs close to the Fermi energy induced by pseudomagnetic fields on wafer-scale semiconductor samples. By measuring the hallmark $\\sqrt{n}$ energy spacing and momentum dependence of the ensuing pseudo-LLs with angle-resolved photoemission spectroscopy (ARPES), and with the aid of model calculations, we confirm their quantum hall nature and extract a pseudomagnetic field strength of B = 41\\,T. This is made possible by the presence of a distribution of triangular nanoprisms underneath the monolayer graphene in our samples based on the well-established platform of epitaxial graphene on SiC substrates\\cite{emtsev_towards_2009,riedl_quasi-free-standing_2009,bostwick_observation_2010,zhou_substrate-induced_2007}, as revealed by a combination of atomic force microscopy (AFM) and scanning tunneling microscopy (STM) measurements.\\par\nOur topographic images of these samples (Fig.\\,\\ref{fig:f2_afm}A inset) exhibit the well-known terraces and step edges of graphene grown on 6H-SiC\\cite{emtsev_towards_2009}, which are due to a miscut of the wafers from the $(0001)$ direction of up to $0.1^\\circ$. A population of triangular-shaped nanoscale features are identified on the terraces of our samples (Fig.\\,\\ref{fig:f2_afm}A), that appear similar to those reported on similar substrates\\cite{bolen_graphene_2009,momeni_pakdehi_homogeneous_2019}. These nanoprisms appear during the growth process of graphene on 6H-SiC and are controllable by the Argon flow in the chamber\\cite{momeni_pakdehi_homogeneous_2019}. They cover between 5\\% and 10\\% of the terraces and are completely covered by monolayer graphene, the latter being demonstrated by our AFM adhesion images (see Supplementary Material Figs.\\,S8 and S9B). They are equilateral, have a narrow size distribution around $300\\,\\textrm{nm}$ side length, are oriented in the same direction, and are about $(2.7\\pm 0.7)\\, \\textrm{\\AA}$ deep (Fig.\\,\\ref{fig:f2_afm}B), which corresponds to a single missing SiC double layer or $\\frac{1}{6}$ of the 6H-SiC unit cell. This leads to a change in the registry between the silicon atoms in the top layer of the substrate and the graphene as illustrated in Fig.\\,\\ref{fig:f2_afm}D. The strain created inside the nanoprisms cannot be relieved, because the nanostructures are continuously covered by monolayer graphene without additional grain boundaries, as corroborated by our STM images across the edge (see Supplementary Material Fig.\\,S9A). To get a more detailed view of the strain pattern we perform additional detailed atomic resolution STM measurements. The images taken inside and outside the nanoprisms (Fig.\\,\\ref{fig:f2_afm}E) show the expected $(6\\sqrt{3}\\times6\\sqrt{3})\\text{R}30^\\circ$ modulation with respect to SiC on top of the carbon honeycomb lattice\\cite{riedl_structural_2010}. However, taking the difference of the two Fourier transformed images (Fig.\\,\\ref{fig:f2_afm}F) reveals a shear strain pattern inside the nanoprism, with a maximal observed strain of roughly $3^\\circ$.\n\n\\begin{figure}\n\\makebox[\\textwidth]{\\includegraphics[width=184mm]{F1_ARPES_v9.png}}\n\\caption{\\textbf{Momentum-resolved visualization of Landau levels.} (\\textbf{A}) ARPES cut through the Dirac cone at the K point at 300\\,K. The data have been divided by the Fermi function and symmetrized to compensate for matrix element effects\\cite{shirley_brillouin-zone-selection_1995}. (\\textbf{B}) Cut along the energy axis integrated around the K point in (\\textbf{A}). (\\textbf{C}) Second derivative of the data in (\\textbf{A})\\cite{zhang_precise_2011}. (\\textbf{D}) Inverted second derivative of the data shown in (\\textbf{B}) after smoothing. (\\textbf{A})--(\\textbf{D}) Landau levels (LLs) are indicated by arrows. (\\textbf{E}) Summary of LL data sets, with model fit according to Eqn.\\,\\ref{eq:landaulevels} shown in black; the 95\\% confidence interval of the fit is shown in grey. Different symbols indicate different samples and temperatures: sample A (6\\,K) [hexagons], sample B (6\\,K) [squares], sample B 2nd data set (6\\,K) [stars], sample B (300\\,K) [diamonds], sample C (6\\,K) [circles], and sample C 2nd data set (6\\,K) [triangles]. ARPES data for the additional samples can be found in the supplementary Fig.\\,S7. Inset: Same data plotted versus $\\sqrt{n}$, giving the expected linear behaviour for LLs in a Dirac material. (\\textbf{F}) Sketch of various mechanisms which may lead to ARPES intensity inside the cone. Neither electron-phonon coupling nor contamination from bilayer graphene can explain the experimental findings.}\n\\label{fig:f1_arpes}\n\\end{figure}\n\nIn order to confirm if the induced strain pattern indeed leads to flat Landau levels close to the Fermi energy, we perform a series of high-resolution ARPES measurements. ARPES is a momentum- and energy-resolved technique that has proven to be a powerful tool in directly studying the electronic band structures of a vast variety of quantum phases of matter, from strongly-correlated electron systems and high-T$_{c}$ superconductors\\cite{damascelli_probing_2004} to topological insulators and semimetals\\cite{hsieh_topological_2008,chen_experimental_2009,lv_observation_2015}. Yet no study of quantum Hall states has been performed, since ARPES is strictly incompatible with the application of magnetic fields, as essential crystal momentum information carried by the photoemitted electrons would be lost through interaction with the field. This however is different for pseudomagnetic fields, as they only interact with the Dirac electrons inside the material. We note that, while a recently developed momentum-resolved technique amenable to magnetic fields has been reported\\cite{jang_full_2017}, it necessarily requires sophisticated heterostructures, physically accessible fields, and is limited to a small sector of the Brillouin zone.\n\nOur ARPES data, which -- due to the $\\sim$1\\,mm spot size of the photon source -- correspond to the spatial average over unstrained and strained regions of the sample, show the expected Dirac cone as well as new flat bands that gradually merge with the linear dispersion (Figs.\\,\\ref{fig:f1_arpes}A and \\ref{fig:f1_arpes}C). The unequal energy spacing of these newly observed bands can be extracted from cuts along the energy direction at the K point (Fig.\\,\\ref{fig:f1_arpes}B) and their second derivative (Fig.\\,\\ref{fig:f1_arpes}D). By plotting the positions of these bands (Fig.\\,\\ref{fig:f1_arpes}E), we observe the distinct $\\sqrt{n}$ energy spacing which is a hallmark of LLs for graphene's massless Dirac charge carriers\\cite{geim_rise_2007}, where $n$ is the integer LL index. The spectrum of LLs in graphene is given by\\cite{castro_neto_electronic_2009}\n\\begin{equation}\n E_n=\\text{sgn}(n)\\sqrt{2v_F^2\\hbar eB\\cdot |n|}+E_{DP}\n \\label{eq:landaulevels}\n\\end{equation} \nwhere $v_F$ is the velocity of the electrons at the Fermi level, $\\hbar$ the reduced Planck constant, $e$ the electron charge, $B$ the magnitude of the (pseudo-)magnetic field, and $E_{DP}$ the binding energy of the Dirac point. Using the ARPES dispersion map in Fig.\\,\\ref{fig:f1_arpes}A, the Fermi velocity is determined to be $v_F=(9.50\\pm0.08)\\times10^5\\ \\textrm{ms}^{-1}$ (see Supplementary Material Fig.\\,S1). Fitting our experimental data to Eqn.\\,\\ref{eq:landaulevels} as done in Fig.\\,\\ref{fig:f1_arpes}E, we extract the magnitude of the pseudomagnetic field, which yields $B=(41\\pm 2)\\ \\textrm{T}$. Remarkably, this pseudomagnetic field value is consistent between several samples from cryogenic temperatures (6\\,K) up to room temperature. The model fit also consistently pinpoints the binding energy of the Dirac point to $E_{DP}=(460\\pm 10)\\ \\textrm{meV}$ relative to the Fermi level, which agrees well with previous reports on this sample system\\cite{bostwick_quasiparticle_2007,emtsev_towards_2009} and is attributed to charge transfer from the SiC substrate to the graphene layer. Additionally, the LLs are only resolved in the upper part of the Dirac cone, closer to the Fermi level\\cite{levy_strain-induced_2010, song_high-resolution_2010}. We attribute this effect to the increased scattering phase space as one moves away from the Fermi level, which manifests itself in our ARPES data by an increased line width of the bands (Supplementary Material Fig.\\,S1).\n\nAs for other alternative explanations of the data, we note that while previous ARPES studies of graphene on SiC have shown a rich variety of features\\cite{ludbrook_evidence_2015,ohta_controlling_2006}, the signature $\\sqrt{n}$ spacing of the levels (Fig.\\,\\ref{fig:f1_arpes}E and inset) allows us to unambiguously distinguish the observed effect from other possibilities (Fig.\\,\\ref{fig:f1_arpes}F). For example, if spectral weight inside the Dirac cone arose from coupling of electrons to phonons\\cite{ludbrook_evidence_2015}, it would be limited to characteristic vibrational energies. Similarly, contributions from bilayer and higher order graphene layers, which can appear in small quantities near step edges of the substrate during the growth process\\cite{ohta_controlling_2006} (see also AFM adhesion image in the Supplementary Material Fig.\\,S9B), would lead to a manifold of bands, but would not reproduce the observed band structure\\cite{ohta_interlayer_2007,marchenko_extremely_2018}. Furthermore, previously reported plasmaronic interactions in samples with higher electronic doping \\cite{bostwick_observation_2010} can also be excluded. They lead to renormalizations of electronic bands around the Dirac point, but show a distinctly different spectrum than what is observed in our experiments. Finally, the effects of different defect geometries in graphene and their influence on the Dirac cone dispersion have recently been discussed \\cite{kot_band_2018}, but do not lead to flat bands around the Dirac point.\n\n\\begin{figure}\n\\makebox[\\textwidth]{\\includegraphics[width=183mm]{F3_THEORY_v8.png}}\n\\caption{\\textbf{Model calculation of strain-induced Landau levels.} (\\textbf{A}) Top: Honeycomb lattice, with the two sublattices A (red) and B (yellow). The black arrows indicate the symmetry of the strain pattern. Bottom: Triangular flake with strain-induced pseudomagnetic field $B = 41$\\,T. The colour scale indicates the relative bond stretching. (\\textbf{B}) Spectral function for the gapless case with Semenoff mass $M = 0\\ \\textrm{meV}$. (\\textbf{C}) Energy cut through the Dirac point (K) of the spectral function in (\\textbf{B}). The dashed grey lines indicate the position of the Landau levels (LL) predicted by Eqn.\\,\\ref{eq:landaulevels}. (\\textbf{D}) Spectral function averaged over a uniform distribution of Semenoff masses $M \\in [-135, 135]$\\,meV. (\\textbf{E}) Energy cut through the Dirac point (K) of the spectral function in (\\textbf{D}). The shaded grey area indicates the broadening of the Landau levels predicted by Eqns.\\,\\ref{eq:landaulevels} and \\ref{eq:landaulevels_mass}.}\n\\label{fig:f3_theory}\n\\end{figure}\n\nTo gain deeper insights on the origin of the observed LLs, we model a region of graphene experiencing a uniform strain-induced pseudomagnetic field. We use the simplest such strain pattern, worked out by Guinea et al.\\cite{guinea_energy_2010}, which exhibits the triangular symmetry of the underlying honeycomb lattice. Using a tight-binding approach, we directly simulate a finite-size strained region with open boundary conditions and armchair edges (further details in Methods section). We find that the observed LL spectra can be well reproduced by a triangular flake of side length $L = 56$\\,nm (Fig.\\,\\ref{fig:f3_theory}A), subject to a uniform pseudomagnetic field $B = 41$\\,T over the entire flake (Fig.\\,\\ref{fig:f3_theory}A). The maximal strain (or relative bond stretching) reaches around $3\\%$, which is in good agreement with our STM measurements. The ARPES data can be simulated by calculating the energy and momentum-resolved spectral function $A(\\bm{k},\\omega)$ of this triangular flake, here shown in Figs.\\,\\ref{fig:f3_theory}B and \\ref{fig:f3_theory}C. Our simulation clearly reproduces the main features of the ARPES data, namely levels that: (i) follow $\\sqrt{n}$ spacing in energy; (ii) are flat \\emph{inside} the Dirac cone and merge with the linearly dispersing bands; (iii) become less clearly resolved with increasing index $n$.\n\nFeatures (ii) and (iii) can be understood by comparing the characteristic size of a Landau orbit $\\propto\\sqrt{n}\\,l_B$ (with the magnetic length $l_B=\\sqrt{\\frac{\\hbar}{eB}}$) to the length scale $\\lambda$ on which the pseudomagnetic field is uniform. For LLs to exist, an electron on a given Landau orbit must experience a uniform pseudomagnetic field\\cite{settnes_pseudomagnetic_2016}, leading to the condition $\\sqrt{n}\\,l_B \\ll \\lambda$. Hence, for large fields $B$ or large $\\lambda$, flat bands are expected across the entire Brillouin zone, whereas Dirac cones are recovered in the opposite limit (see Supplementary Material Fig.\\,S2). The bands observed in the ARPES data can thus be understood as LLs, where the orbit size is only somewhat smaller than $\\lambda$: by comparing the experimental data and the model calculation, we estimate $l_B \\sim 4$~nm and $\\lambda \\sim 30$\\,nm (see Methods section). Furthermore, since the size of Landau orbits grows as $ \\sim \\sqrt{|n|}$, eventually it becomes comparable to $\\lambda$, explaining why levels with higher index $n$ are less clearly resolved.\n\nHowever, our simple model (Figs.\\,\\ref{fig:f3_theory}B and \\ref{fig:f3_theory}C) consistently exhibits a sharp zeroth Landau level (LL0), which is absent in the ARPES data. This discrepancy is surprising, since LL0 is known to be stable against inhomogeneities of the magnetic field as well as against disorder, as long as the latter preserves the chiral symmetry of graphene\\cite{aharonov_ground_1979}. Below, we provide a possible mechanism that broadens LL0 without substantially affecting the higher LLs. It has been argued that graphene grown on SiC is subject to a sublattice-symmetry-breaking potential arising from the interaction with the substrate\\cite{zhou_substrate-induced_2007}. The minimal theoretical model describing this effect, which acts as a staggered potential between sublattices A and B, is the so-called Semenoff mass $M$\\cite{semenoff_condensed-matter_1984}. This mass term opens a gap at the Dirac point and shifts the LL spectrum for $n\\neq0$ to (for a more detailed discussion, and the particular case of $n=0$, see Supplementary Material)\\cite{hunt_massive_2013}:\n\\begin{equation}\n E_n = \\text{sgn}(n) \\sqrt{ 2 v_F^2 \\hbar e B\\cdot|n| + M^2}+E_{DP}.\n \\label{eq:landaulevels_mass}\n\\end{equation}\nHowever, a \\emph{uniform} mass term $M$ cannot explain the ARPES data. Indeed, a fit of the observed LL spectrum to Eqn.\\,\\ref{eq:landaulevels_mass} returns $M=(150\\pm 5)\\ \\textrm{meV}$, but places the Dirac point at an unrealistic binding energy of $E_{DP}=390\\ \\textrm{meV}$ (see Supplementary Material Fig.\\,S3 and Fig.\\,S6). Therefore, we postulate that the mass term $M$ varies on a length scale much greater than the magnetic length $l_B \\sim$\\,4\\,nm, but smaller than the ARPES spot size ($\\sim$\\,1\\,mm). In that situation, our ARPES measurements would simply average over spectral functions described by different mass terms. This is shown in Figs.\\,\\ref{fig:f3_theory}D and \\ref{fig:f3_theory}E for a uniform distribution in the interval $M \\in [-135, 135]$\\,meV. As evident from Eqn.\\,\\ref{eq:landaulevels_mass}, the distribution of mass terms affects LL0 most, while merely contributing an additional broadening to the higher levels. Note that, as observed experimentally, the variation of the mass term is not limited to the strained areas, but instead is a property of the whole sample; as a result, ARPES always picks up a spatial average of strained areas with LLs and unstrained areas with the usual Dirac cone dispersion, both having the same distribution of mass terms and corresponding Dirac point gaps. This phenomenological model is in good agreement with the experimental data and may renew interest in the variation of the mass term in this sample system\\cite{zhou_substrate-induced_2007}.\n\n\\section*{Discussion}\n\nThis study provides the first demonstration of the room temperature strain-induced quantum Hall effect in graphene on a wafer-scale platform, as well as the first direct momentum-space visualization of graphene electrons in the strain-induced quantum Hall phase by ARPES, whereby the linear Dirac dispersion collapses into a ladder of quantized LLs. This opens a path for future momentum-resolved studies of strain-induced, room temperature-stable topological phases in a range of materials including Dirac and Weyl semimetals\\cite{pikulin_chiral_2016,cortijo_elastic_2015,liu_quantum_2017}, monolayer transition metal dichalcogenides\\cite{rostami_theory_2015}, and even nodal superconductors\\cite{massarelli_pseudo-landau_2017,nica_landau_2018}, all under large, potentially controllable pseudomagnetic fields. Importantly, these systems will feature time reversal invariant ground states -- otherwise impossible with a true magnetic field -- and may act as future building blocks for pseudo spin- or valley-tronic based technologies\\cite{low_strain-induced_2010}. In light of the recently discovered unconventional superconductivity in 'magic angle' bilayer graphene\\cite{cao_correlated_2018,cao_unconventional_2018}, strain-induced pseudomagnetic fields likewise raise the possibility of engineering exotic variants of correlated states including superconductivity in LLs\\cite{uchoa_superconducting_2013} and fractional topological phases\\cite{ghaemi_fractional_2012}. Our results lay the foundations for bottom-up strain-engineering of novel quantum phases at room temperature and on a technologically relevant wafer-scale platform.\n\n\\section*{Materials and Methods}\n\n\\subsection*{Sample growth and characterization}\n\nGraphene samples with a carbon buffer layer were epitaxially grown on commercial 6H-SiC substrates. The substrates were hydrogen-etched prior to the growth under argon atmosphere. Details are described by \\emph{S. Forti and U. Starke}\\cite{forti_epitaxial_2014}. AFM characterization measurements were taken at the Max Planck Institute in Stuttgart. Adhesion images correspond to the force necessary to retract the tip from the sample. Adhesion is sensitive to the graphene coverage on the sample and can thus distinguish between zero layer, monolayer and bilayer graphene with sensitivity to grain boundaries.\n\n\\subsection*{ARPES measurements}\n\nExperiments were performed at UBC in a ultra-high vacuum chamber equipped with a SPECS Phoibos 150 analyzer with $\\Delta E=6\\,\\textrm{meV}$ and $\\Delta k=0.01\\,\\textrm{\\AA}$ optimum energy and momentum resolutions, respectively, at a base pressure of better than $p=7\\times10^{-11}\\,\\textrm{Torr}$. Photons with an energy of $21.2$\\,eV were provided by a SPECS UVS300 monochromatized gas discharge lamp. Our homebuilt six-axis cryogenic manipulator allows for measurements between 300\\,K and 3.5\\,K. Additional data sets were taken at UBC with a second ARPES setup equipped with a Scienta R4000 analyzer and a Scienta VUV5000 UV source with $\\Delta E=1.5\\,\\textrm{meV}$ and $\\Delta k=0.01\\,\\textrm{\\AA}^{-1}$ optimum energy and momentum resolutions, respectively, for $21.2\\ \\textrm{eV}$ photons. The samples were annealed at 600$^\\circ$C for about 2\\,h at $p=1\\times10^{-9}\\,\\textrm{Torr}$ and then at 500$^\\circ$C for about 10\\,h at $p=5\\times10^{-10}\\,\\textrm{Torr}$ immediately before the ARPES measurements.\n\n\\subsection*{STM measurements}\n\nExperiments were performed at UBC under ultra-high vacuum conditions ($<5\\times10^{-12}$\\,mbar) using a low-temperature scanning tunnelling microscope (Scienta Omicron) at liquid helium temperatures ($\\sim$\\,4.2\\,K). All images were acquired in constant-current mode using a cut\\linebreak platinum-iridium tip, which was conditioned by voltage pulsing and gentle indentation into a Ag(111) crystal. The samples were annealed at 550$^\\circ$C overnight with a final pressure of $p=3\\times10^{-10}\\,\\textrm{mbar}$ \\emph{in situ} prior to the STM measurements.\n\n\\subsection*{Model calculation}\n\nWe considered a minimal tight-binding model on the honeycomb lattice with nearest-neighbour hoppings and a sublattice-symmetry breaking Semenoff\\cite{semenoff_condensed-matter_1984} mass term $M$:\n\\begin{equation}\nH = -t \\sum_{<\\bm{r}, \\bm{r'}>} \\left( c_A^\\dagger(\\bm{r)} c_B(\\bm{r'}) + \\text{H.c.} \\right) + M \\left( \\sum_{\\bm{r}} c_A^\\dagger(\\bm{r}) c_A(\\bm{r}) - \\sum_{\\bm{r'}} c_B^\\dagger(\\bm{r'}) c_B(\\bm{r'}) \\right)\n\\label{eq:tight_binding}\n\\end{equation}\nwhere $c^\\dagger_A(\\bm{r})$ ($c^\\dagger_B(\\bm{r'})$) creates an electron in the $p_z$ orbital at lattice site $\\bm{r}$ ($\\bm{r'}$) on the sublattice A (B) of the honeycomb lattice, $t = 2.7\\ \\text{eV}$ and the nearest-neighbour distance is $a_0 =0.142$\\,nm. We neglected the electron spin, and thus considered effectively spinless fermions.\n\nWe constructed a flake in the shape of an equilateral triangle of side length $L \\sim 56$\\,nm. The use of armchair edges avoids the zero-energy edge modes appearing for zigzag edges\\cite{castro_neto_electronic_2009}. We applied the simplest strain pattern respecting the triangular symmetry of the problem at hand, namely, the pattern introduced by \\textit{Guinea et al.}\\cite{guinea_energy_2010} which gives rise to a uniform (out-of-plane) pseudomagnetic field\n\\begin{equation}\n \\bm{B} = 4 u_0 \\frac{\\hbar \\beta}{e a_0} \\hat{\\mathbf{z}}\n\\end{equation}\nwhere $\\beta \\approx 3.37$ in graphene\\cite{settnes_pseudomagnetic_2016}, and the corresponding displacement field is given by\n\\begin{equation}\n \\mathbf{u}(r,\\theta) = \n \\begin{pmatrix}\n u_r \\\\\n u_\\theta\n \\end{pmatrix} \n =\n \\begin{pmatrix}\n u_0 r^2 \\sin(3\\theta) \\\\\n u_0 r^2 \\cos(3\\theta)\n \\end{pmatrix}.\n\\end{equation}\nThe hopping parameter renormalization induced by this displacement field is calculated using the simple prescription:\n\\begin{align}\nt \\rightarrow t_{ij} = t \\exp \\left[ -\\frac{\\beta}{a_0^2} \\left( \\epsilon_{xx} x_{ij}^2 + \\epsilon_{yy} y_{ij}^2 + 2 \\epsilon_{xy} x_{ij} y_{ij} \\right) \\right]\n\\label{eq_hoppings_strain}\n\\end{align}\nwhere $(x_{ij}, y_{ij}) \\equiv \\bm{r}_i - \\bm{r}_j$ is the vector joining the original (unstrained) sites $i$ and $j$, and\n\\begin{equation}\n \\epsilon_{ij} = \\frac{1}{2} \\left[ \\partial_j u_i + \\partial_i u_j \\right]\n\\end{equation}\nis the strain tensor corresponding to the (in-plane) displacement field $\\mathbf{u}$. Outside the strained region (which we take as a triangle of slightly smaller length $L_S \\sim 48$\\,nm), we allowed the strain tensor to relax: $\\bm{\\epsilon} \\rightarrow e^{-\\frac{r^2}{2\\sigma^2}} \\bm{\\epsilon}$, where $r$ is the perpendicular distance to the boundary of the strained region, and $\\sigma \\sim 1$\\,nm. We defined the length scale of the homogeneous magnetic field $\\bm{B}$ to be the diameter of the largest inscribed circle in the triangle of side $L_S$: $\\lambda \\equiv L_S\/\\sqrt{3} \\sim 28$\\,nm. We stress here that our simulated flakes are much smaller than the experimentally observed triangular features of size $\\sim 300$\\,nm. The fact that we nevertheless reproduce the experimental features underlines how the number of observable LLs is limited by the length scale of the homogeneous pseudomagnetic field $\\lambda$, rather than by the size $L$ of the nanoprisms themselves. This length scale could be caused by the more complicated strain pattern present in the nanoprisms or be induced by disorder.\n\nWe then diagonalized the Hamiltonian (Eqn.\\,\\ref{eq:tight_binding}) with hopping parameters given by Eqn.\\,\\ref{eq_hoppings_strain} to obtain the full set of eigenstates $\\ket{n}$ with energies $E_n$, and computed the momentum-resolved, retarded Green's function using the Lehman representation\n\\begin{align}\n G_\\alpha^R(\\bm{k},\\omega) &= \\sum_n \\frac{ | \\bra n c^\\dagger_\\alpha({\\bm{k}}) \\ket{0} |^2}{\\omega - (E_n - E_0) -i \\eta}\n\\end{align}\nwhere $\\alpha = A,B$ is a sublattice (band) index, and $\\eta \\sim 20$\\,meV is a small broadening parameter comparable to the experimental resolution. \nWe then compute the one-particle spectral function,\n\\begin{align}\nA(\\bm{k}, \\omega) = -\\frac{1}{\\pi} \\sum_\\alpha \\text{Im}\\left[ G_\\alpha^R(\\bm{k}, \\omega)\\right] \n\\end{align}\nwhich is proportional to the intensity measured in ARPES (modulo the Fermi-Dirac distribution and dipole matrix elements). We note that using a finite system introduces two main effects in the momentum-resolved spectral function: the appearance of a small finite-size gap at the Dirac points (in the absence of a magnetic field) and a momentum broadening of the bands.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nUnderstanding the origin of the Ultra High Energy Cosmic Rays (UHECR) with energy above $\\sim 5 \\; 10^{19}$ eV\nconstitute a real challenge for theoretical models, since their acceleration requires extreme conditions hardly fulfilled by known astrophysical objects.\nPrevious studies considered a number of potential sources, including gamma ray bursts, active galactic nuclei, large scale jets and neutron stars, but results are still inconclusive \\cite{reviews}. \n\nIf, as many believe, UHECRs have an extragalactic origin, a suppression, known as the Greisen-Zatsepin-Kuzmin (GZK) cutoff, should be observed in the spectrum at an energy\n$\\sim 7 \\; 10^{19} {\\rm eV}$\n\\cite{GZK}.\nThis is because during propagation energetic protons undergo $p\\gamma$ interactions (mainly photo-pion production) in the\nCosmic Microwave Background (CMB)\nwith an energy loss length which decreases dramatically from $\\sim 650$ to $\\sim 20 \\, {\\rm Mpc}$ between energies of $7 \\; 10^{19}$ to $3 \\; 10^{20} {\\rm eV}$ \\cite{pgamma}.\nThe detection of a number of events above $7 \\, 10^{19}$ eV\nraised interest in non-acceleration\nmodels\nthat\ninvolve new physics and do not require the existence of a cutoff in the spectrum \\cite{reviews}.\nHowever, the issue of the presence\nof the GZK feature\nis still debated \\cite{daniel}.\nIn the following we assume that astrophysical objects capable of accelerating UHECRs\ndo exist and we discuss possible ways to identify them.\n\nIf UHECRs are\nnot significantly \ndeflected by the\nintergalactic magnetic field (IGMF)\ntheir arrival direction should point back to the position of their accelerators.\nIn principle, this fact can\nbe used to identify accelerators located at distances smaller than the proton loss length, that constitutes a sort of horizon for CRs. However, even in this case, the small event statistics and the high uncertainty\nin the determination of the CR arrival direction\nwould make any identification problematic. \nThe better angular resolution and\ncollection area of AUGER will hopefully provide \nadequate data to address this issue.\n\n\n\nThe interactions between UHECRs and CMB photons generate secondary gamma rays and electron-positron pairs (hereafter referred to as {\\it electrons}) that in turn initiate an electromagnetic (EM) cascade in the universal photon background \\cite{cascade}.\nSuch a cascade would appear to a distant observer as a flux of GeV\/TeV photons.\nRecently, Ferrigno et al. \\cite{carlo} suggested to\nidentify the sources of UHECRs\nby searching for this radiation.\nTheir calculations show that, in an unmagnetized Universe ($B_{IGMF} = 0$ G), a steady source emitting isotropically $2 \\; 10^{43}$ erg\/s in form of UHECRs ($E > 10^{19}$ eV) can be detected by a Cherenkov telescope like HESS up to a distance of $\\sim 100$ Mpc.\nIn this\ncase the source would be point-like, because in the absence of IGMF the EM cascade is one-dimensional and propagates radially away from the accelerator. \nUnfortunately, the scenario changes dramatically if\nthe more realistic case of a\nmagnetized Universe is considered. This is because low energy electrons produced during the last steps of the cascade are effectively deflected and eventually isotropized if their Larmor radius is smaller or comparable with the Compton cooling length. This condition is satisfied when the IGMF is above\n$\nB_{iso} \\sim 10^{-12} (E_{\\gamma}\/TeV) {\\rm G}\n$,\n$E_{\\gamma}$ being the energy of the Compton photon. This implies that \\textit{unless the IGMF is extremely weak} ($<< B_{iso}$), \nelectrons emit Compton photons after being fully isotropized. Thus, \\textit{an extended halo of emitting pairs forms around UHECR sources}. The radiation from the halo is emitted isotropically, resulting in a \\textit{very extended, and thus hard to be detected, gamma ray source} \\cite{cascade}.\n\nHowever, if the IGMF close to the accelerator is at the ${\\rm nG}$ level, first generation electrons ($E \\sim 10^{19}$ eV) generated during $p\\gamma$ interactions cool rapidly by emitting ${\\rm GeV}$ synchrotron photons and the development of the cascade is strongly inhibited. \nWe propose the possibility to detect these synchrotron photons from UHECR sources \\cite{noi}. \nThis is of great interest because of the following reasons.\nBoth synchrotron emitting electrons and parent protons are extremely energetic and not appreciably deflected by the IGMF, at least on the first Mpcs distance scale.\nFor this reason, \\textit{synchrotron photons are emitted in the same direction of parent protons}. Thus, they move away from the source almost radially, and \\textit{the observed radiation is expected to be point-like}, and thus easily detectable and distinguishable from the extended cascade component.\nRemarkably, a detection of these sources would allow to infer the value of the IGMF close to the accelerator, constraining it in the range $\\approx 10^{-10} \\div 10^{-8}$ G.\nFinally, since the Universe is transparent to ${\\rm GeV}$ photons, powerful UHECR accelerators located outside the CR horizon might be identified in this way. \n\n\\section{Development of the electromagnetic cascade}\n\nIn this section we describe the development of the EM cascade initiated by a UHECR.\nConsider a proton with energy $E_{p,20} = E_p\/10^{20}$ eV. The typical energies of photons and electrons produced in $p\\gamma$ interactions are $\\sim 10^{19} E_{p,20}$ and $\\sim 5 \\; 10^{18} E_{p,20}$ eV respectively \\cite{pgamma}.\nIn the absence of IGMF, such electrons and photons interact via Compton and pair production processes with photons in the CMB and radio background.\nIn general, this would lead to the development of an EM cascade, in which the number of electrons and photons increases rapidly.\nIn fact, due to the extremely high energy of the particles considered here, each interaction occurs in the limit $\\Gamma = \\epsilon_b E \\gg 1$, where $\\epsilon_b$ and $E$ are the energies of the background photon and of the energetic electron (photon) respectively, both calculated in units of the electron rest mass energy.\nUnder this condition the Compton scattering is in the extreme Klein-Nishina limit, namely, the upscattered photon carries away most of the energy of the incoming electron.\nThe same happens during a pair production event, in which most of the energy goes to one of the two outgoing electrons.\nTherefore the problem reduces essentially to a single-particle problem, in which a leading particle loses continuously energy and changes state from electron to photon and back\ndue to alternate Compton\/pair production interactions.\nThus, the effective loss length of an energetic electron can be identified with the loss length of the leading particle \\cite{gould}. \nWhen the leading particle loses its energy until $\\Gamma \\approx 1$,\nthe cascade enters the particle multiplication phase, in which the particle energy is roughly divided in half in every collision. This phase ends up with a large number of low energy electrons and photons.\nOn the other hand, if a IGMF is present, electrons also lose energy via synchrotron emission, subtracting energy to the cascade.\nIn Fig.~\\ref{eloss} we show the effective electron loss length for Compton\/pair production (solid), together with the synchrotron loss length for a IGMF equal to $0.1$, $1$ and $10 ~ {\\rm nG}$ (dashed lines).\nIt can be seen that if the IGMF is at the level of 1 nG or more,\nall the electrons \nwith energy above $\\sim 10^{18}$ eV cool fast via synchrotron losses and the development of the cascade is strongly suppressed.\n\nSummarizing, three different regimes, corresponding to different values of the IGMF, can be distinguished:\n\n\\begin{itemize}\n\\item[$\\bullet$]{{\\bf Regime I:} $B_{IGMF} \\ll B_{iso} \\sim 10^{-12}$ G. The EM cascade is not affected at all by the\nIGMF.}\n\\item[$\\bullet$]{{\\bf Regime II:} $B_{iso} \\le B_{IGMF} \\ll B_{syn} \\sim 10^{-9}$ G. No energy is subtracted to the EM cascade due to synchrotron losses, but low energy electrons are effectively isotropized by the IGMF.}\n\\item[$\\bullet$]{{\\bf Regime III:} $B_{IGMF} \\ge B_{syn}$. The development of the EM cascade is strongly suppressed since its very first steps due to strong synchrotron losses.}\n\\end{itemize}\n\n\\section{Regime I: one-dimensional cascade}\n\nIf the IGMF strength is much less than $B_{iso} \\sim 10^{-12}$ G, electrons in the cascade do not suffer synchrotron losses, nor are they deflected. Thus, the EM cascade develops along a straight line. In this case, the calculations by Ferrigno et al. apply, and nearby and powerful UHECR sources might be detected as point like TeV sources by currently operational Cherenkov telescopes \\cite{carlo}.\n\nIn principle, since the strength of the large scale IGMF is basically unknown \\cite{Bfield}, such a low values of the field cannot be ruled out. However, this is probably not a good assumption in the vicinity of UHECR accelerators, where the IGMF is expected to be appreciable, especially if such accelerators are, as it seems reasonable to believe, correlated with the structures in the Universe. \nThe cascade might still be one-dimensional if its last steps develop sufficiently far away from the source, in a region of very low IGMF. Another necessary condition is that the IGMF close to the source must be small enough ($\\ll 10^{-9}$ G) to avoid a suppression of the cascade due to synchrotron losses of first generation electrons. \n\n\\section{Regime II: extended pair halos}\n\nIf the IGMF is strong enough to deflect the electrons in the cascade, but not enough to make synchrotron losses relevant (namely, $10^{-12} {\\rm G} \\le B_{IGMF} \\ll 10^{-9}$ G), then the EM cascade fully develops, low energy electrons are isotropized, and a very extended pair halo forms around the UHECR source. For an isotropic source, the size of the halo can be roughly estimated as follows. Let $E_{\\gamma}^{obs}$ be the energy of the gamma ray photons observed from the Earth. Such photons are CMB photons Compton-upscattered by electrons with energy $E_e \\sim 20 (E_{\\gamma}^{obs}\/{\\rm TeV})^{1\/2}$ TeV. These are the electrons forming the pair halo. \nSince electrons are rapidly isotropized in the IGMF, one can assume that they do not propagate away from the sites in which they are created.\nElectrons in the halo are in turn produced by parent photons with energy $E_{\\gamma}^{par} \\lesssim E_e$. \nSince the photon mean free path against pair production in the infrared background $\\lambda_{pp}$ decreases rapidly with increasing energy \\cite{IRabs}, we can safely neglect the contribution to the halo size from older generation (higher energy) photons. \nThus, the size of the halo \ncan be roughly estimated as $l_{halo} \\sim \\lambda_{pp}(E_{\\gamma}^{par})$ \\cite{cascade}.\nFor a $\\sim 20$ TeV photon the mean free path is about a few tens of megaparsecs (see Fig. 2 in \\cite{felixICRC}). In fact, for the situation considered here, the size of the halo is even bigger, since the UHECR protons and the first generation electrons\npropagate $\\sim 10 \\div 20$ Mpc before\ninitiating the EM cascade (see Fig. 2).\n Thus, a conservative estimate of the apparent angular size of the halo at 1 TeV can be given by: $\\vartheta \\sim (l_{halo}\/D) \\approx 10^o (l_{halo}\/20Mpc) (D\/100Mpc)^{-1}$, where $D$ is the distance of the source.\nThis indicates that \\textit{pair halos are extremely extended}, bigger than the field of view of Cherenkov telescopes (the HESS field of view is $\\sim 5^o$) \\textit{and thus hardly detectable}. The problem becomes even worse if one considers also protons with energy below $10^{20}$ eV, which have much longer loss length, increasing up to $\\sim 1$ Gpc for proton energies equal to $\\sim 5 \\; 10^{19}$ eV. On the other hand, such protons are likely to contribute only to the TeV flux of very distant sources.\n\nIn the recent work by Armengaud et al. \\cite{sigl} the deflection of electrons in the IGMF has been neglected, even when a IGMF stronger than $B_{iso} \\sim 10^{-12}$ G was assumed. On the other hand, the authors considered the deflection of $\\sim 10^{20}$ eV protons, which is in fact totally negligible if compared with the full isotropization of electrons. \nTherefore their claim\n about the detectability of cascade gamma-rays \nseems to us over-optimistic.\n\nThe cascade emission peaks at TeV energies \\cite{carlo}, making a detection by GLAST problematic.\n\n\\section{Regime III: synchrotron gamma rays}\n\nIf the IGMF close to the UHECR accelerator is at the level of 1 nG or above, the development of the cascade is strongly suppressed, since the very high energy electrons produced during $p\\gamma$ interactions cool rapidly via synchrotron losses before undergoing Compton scattering. It is evident from Fig. 2 that for a $\\sim nG$ IGMF this is true for electron energies well in excess of $E_e \\sim 10^{18}$ eV. Such electrons emit synchrotron photons with energy \n$E_{syn} \\approx 2 \\, (B\/{\\rm nG}) (E\/10^{19}{\\rm eV})^2 {\\rm GeV}$, detectable by GLAST.\nIt is important to stress that our results are sensitive only to the value of the IGMF {\\it close} to the source, while they are unaffected by the value of the field on much larger scales. \nThis is because synchrotron emitting electrons are produced within a proton interaction length $l_{p\\gamma} \\approx 10$ Mpc from the accelerator.\nAs a consequence, the only assumption required\nis that the size of the magnetized region surrounding the accelerator must be greater or comparable with $l_{p\\gamma}$\nSuperclusters of galaxies constitute an example of large and magnetized regions satisfying our requirement \\cite{Bfield}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{gabiciposter_fig1.ps}\n\\caption{\\label{eloss}Effective electron loss length. Solid: Compton\/pair-production losses in the CMB and radio background (with a cutoff at $2 {\\rm MHz}$ \\cite{radiobackground}). Dashed: synchrotron losses.\n}\n\\end{figure}\n\n\nWe now estimate the angular size of the synchrotron emission.\nAfter propagating over an interaction length, a proton of energy $E_p$ is deflected by an angle\n$\\vartheta_p \\approx 0.8^o (10^{20}{\\rm eV}\/E_p) (B\/{\\rm nG}) \\sqrt{(l_{p\\gamma}\/10{\\rm Mpc})} \\sqrt{(l_c\/{\\rm Mpc})}$ \nwhere $l_c$ is the IGMF coherence length \\cite{deflection}.\nDue to the high energies considered, secondary electrons produced in $p\\gamma$ interactions move in the same direction of the parent protons.\nIn a cooling time electrons are deflected by an angle $\\vartheta_e \\sim \\alpha \\lambda \/ R_L$,\n$R_L$ being the electron Larmor radius and $\\alpha$ a number of order unity representing the probability that the leading particle is actually an electron \\cite{gould}.\nRemarkably, if expressed as a function of the synchrotron photon energy, \nthe deflection angle is independent on the magnetic field strength and reads: $\\vartheta_e \\approx 0.5^o (E_{syn}\/10 GeV)^{-1}$.\nThus, an observer at a distance $D$ would see a source with angular size: \n$\\vartheta_{obs} \\approx \\sqrt{\\vartheta_p^2+\\vartheta_e^2} \\left(\\frac{l_{p\\gamma}}{D}\\right)$ that, for $D = 100$ Mpc and for photon energies of $1 \\div 10$ GeV is of the order of a fraction of a degree.\nThis is comparable with the angular resolution of GLAST, that would classify these sources as point-like if they are\nlocated at a distance of $\\sim 100 \\; {\\rm Mpc}$ or more. \nThis leads to the important conclusion that, \\textit{even if synchrotron photons are produced in an extended region of size $\\sim l_{p\\gamma}$ surrounding the accelerator, the resulting gamma ray source would appear point-like to a distant observer}.\n\nIf the proton spectrum extends well above $10^{20}$ eV, these sources might also be detected by \nImaging Atmospheric Cherenkov Telescope arrays (IACT) operating at energies above $100 \\; {\\rm GeV}$. The angular resolution of these instruments is a few arcminutes, and thus the sources will appear extended. However, it has been proven that IACT are powerful instruments to image extended gamma ray sources \\cite{SNR}, and thus they might still detect and map the emission from UHECR sources. Finally, powerful UHECR accelerators located at a distance of 1 Gpc or more would appear as point sources. In the next section we discuss the energy requirement for a detection.\n\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{gabiciposter_fig2.ps}\n\\caption{\\label{spectra}Spectra for a source \nlocated at\n$100{\\rm Mpc}$.\nThe luminosity in UHECRs is $2 \\; 10^{44}{\\rm erg\/s}$, with spectral index is $\\delta = 2$. TOP: $B_{IGMF} = 0.5$(curve 1), $5$ (2), $50{\\rm nG}$(3), $E_{cut} = 10^{21} {\\rm eV}$. BOTTOM: $E_{cut} = 5 \\, 10^{20}$, $10^{21}$, $5 \\, 10^{21} {\\rm eV}$, $B_{IGMF} = 1{\\rm nG}$. Dotted: intrinsic spectra. Solid: spectra after absorption.}\n\\end{figure}\n\n\\section{Detectability and energetics\n\nFig.~\\ref{spectra} shows synchrotron spectra\nfor a steady source\nat a distance of $100 ~ {\\rm Mpc}$ in a uniformly magnetized region of size 20 Mpc.\nSince the radiation is produced in a region about one interaction length away from the source, the actual structure of the field is not crucial.\nSteady state proton and electron spectra have been calculated taking into account all the relevant energy losses and proton escape from the magnetized region.\nSolid lines have been computed \ntaking into account the opacity of the Universe to very high energy photons due to pair production in the cosmic infrared background \\cite{IRabs},\nwhile dotted lines show the unabsorbed spectra.\nThe total luminosity in UHECRs with energy above $10^{19} {\\rm eV}$ is $L_{UHE} = 2 ~ 10^{44} {\\rm erg\/s}$, with a differential energy distribution $\\propto E^{-\\delta} exp(-E\/E_{cut})$, with \n$\\delta =2$.\nResults are quite insensitive to the slope of the spectrum.\n\nIn the top panel of Fig.~\\ref{spectra} $E_{cut} = 10^{21} {\\rm eV}$\nand the IGMF\nis equal to $0.5$, $5$ and $50 \\; {\\rm nG}$ (curves $1$, $2$ and $3$).\nIf the IGMF is significantly greater than $\\sim 50 \\; {\\rm nG}$, the peak of the emission falls at TeV energies, where absorption is very strong.\nOn the other hand, if the field is well below $\\sim 0.5 \\; {\\rm nG}$, synchrotron emission becomes unimportant and the cascade contribution dominates.\nHowever, \\textit{for the broad interval of values of the IGMF strength between $0.5$ and $50 \\; {\\rm nG}$, the formation of a synchrotron point-like gamma ray source seems to be unavoidable}. \n\nIn the bottom panel of Fig.~\\ref{spectra}, our predictions are compared with the sensitivities of GLAST and of a generic\nIACT such as HESS or VERITAS. \nA IGMF of $1 \\; {\\rm nG}$ is assumed and the different curves refer to values of the cutoff energy in the proton spectrum equal to $5 \\; 10^{20}$, $10^{21}$ and $5 \\; 10^{21} {\\rm eV}$ (top to bottom).\nFor such a field, the condition for the detectability of a point source by GLAST is roughly $L_{UHE} \\ge 8 \\; 10^{43} \\div 2 \\; 10^{44} (D\/100{\\rm Mpc})^2 {\\rm erg\/s}$ for $\\delta = 2.0 \\div 2.6$.\nIn contrast, for IACT arrays the minimum detectable luminosity is roughly 2 orders of magnitude higher, since the source has to be located at a distance of $\\sim 1 {\\rm Gpc}$ in order to appear point-like. However, less powerful accelerators can still be detected as extended sources.\nSince the peak of the emission falls at $\\sim 10 ~ {\\rm GeV}$, future IACT operating in the energy range $10 \\div 100$ GeV would be powerful tools to search for these sources. \n\nIf the CR spectrum smoothly \nextends down to GeV energies with slope $\\delta = 2$, the required total CR luminosity for a source to be detected by GLAST is $L_{CR} \\ge 5 \\, 10^{44} (D\/100{\\rm Mpc})^2 {\\rm erg\/s}$.\nFor a beamed source, the required luminosity is reduced by a factor $f_b \\sim 0.02 (\\vartheta_b\/10^o)$, $\\vartheta_b$ being the beaming angle,\nand the detectability condition reads: $L_{CR} \\ge 10^{43} (f_b\/0.02) (D\/100{\\rm Mpc})^2 {\\rm erg\/s}$.\nThis luminosity is small if compared, for example, with the\npower of an AGN jet, that can be as high as\n$10^{47} {\\rm erg\/s}$ \\cite{ghisella}.\nThus,\nastrophysical objects that can in principle satisfy the energy requirement for a detection do exist.\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n~~~~~The purpose of this paper is to bring a new tool in the\nstudy of the spectral type of rank one transformations. Rank one\ntransformations have simple spectrum and in \\cite{Ornstein} D.S.\nOrnstein, using a random procedure, produced a family of mixing\nrank one transformations. It follows that the Ornstein's class of\ntransformations may possibly contain a candidate for Banach's well-known\nproblem whether there exists a dynamical system $(\\Omega,{\\mathcal\n{A}},\\mu,T)$ with simple Lebesgue spectrum. But, in 1993, J.\nBourgain in \\cite{Bourgain} proved that almost surely Ornstein's\ntransformations have singular spectrum. Subsequently, using the\nsame method, I. Klemes \\cite{Klemes1} and I. Klemes \\& K. Reinhold\n \\cite{Klemes2} obtain that mixing \nstaircase transformations of Adams \\cite{Adams1} and Adams \\&\nFriedman \\cite{Adams2} have singular spectrum. They conjecture\nthat all rank one transformations have singular spectrum.\\par\n\nHere we shall exhibit a new class of rank one transformations with\nsingular spectrum. Our assumption include some new class of\nOrnstein transformations and a class of Creutz-Silva rank one\ntransformations \\cite{Creutz-silva}. Our proof is based on\ntechniques introduced by J. Bourgain\n\\cite{Bourgain} in the context of rank one transformations and\ndeveloped by Klemes \\cite{Klemes1}, Klemes-Rienhold\n\\cite{Klemes2}, Dooley-Eigen \\cite{Eigen}, together with some ideas from the proof of \nthe central limit theorem for trignometric sums. The fundamental key, as noted by Klemes\n\\cite{Klemes1}, is the estimation of the $L^1$-norm of a certain\n trigonometric polynomial ${{(|P_m|^2-1)}}$\n. We shall use the method of central limit theorem for trignometric sums to produce an\nestimate of this $L^1$-norm.\n\n\\section*{2. Rank One Transformation by Construction}\n\nUsing the cutting and stacking method described in [Fr1], [Fr2],\none can construct inductively a family of measure preserving\ntransformations, called rank one transformations, as follows\n\\vskip 0.1cm Let $B_0$ be the unit interval equipped with the\nLebesgue measure. At stage one we divide $B_0$ into $p_0$ equal\nparts, add spacers and form a stack of height $h_{1}$ in the usual\nfashion. At the $k^{th}$ stage we divide the stack obtained at the\n$(k-1)^{th}$ stage into $p_{k-1}$ equal columns, add spacers and\nobtain a new stack of height $h_{k}$. If during the $k^{th}$ stage\nof our construction the number of spacers put above the $j^{th}$\ncolumn of the $(k-1)^{th}$ stack is $a^{(k-1)}_{j}$, $ 0 \\leq\na^{(k-1)}_{j} < \\infty$, $1\\leq j \\leq p_{k}$, then we have\n\n$$h_{k} = p_{k-1}h_{k-1} + \\sum_{j=1}^{p_{k-1}}a_{j}^{(k-1)}.$$\n\\vskip 3 cm \\hskip 3.5cm\n \\tower\n\\vskip 3.0cm\n\\noindent{}Proceeding in this way we get a rank one transformation\n$T$ on a certain measure space $(X,{\\mathcal B} ,\\nu)$ which may\nbe finite or\n$\\sigma-$finite depending on the number of spacers added. \\\\\n\\noindent{} The construction of a rank one transformation thus\nneeds two parameters, $(p_k)_{k=0}^\\infty$ (parameter of cutting\nand stacking), and $((a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty$\n(parameter of spacers). Put\n\n$$T \\stackrel {def}= T_{(p_k, (a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty}$$\n\n\\noindent In \\cite{Nadkarni1} and \\cite{Klemes2} it is proved that\nthe spectral type of this transformation is given (upto possibly some discrete measure) by\n\n\\begin{eqnarray}\\label{eqn:type1}\nd\\sigma =W^{*}\\lim \\prod_{k=1}^n\\left| P_k\\right| ^2d\\lambda,\n\\end{eqnarray}\n\\noindent{}where\n\\begin{eqnarray*}\n&&P_k(z)=\\frac 1{\\sqrt{p_k}}\\left(\n\\sum_{j=0}^{p_k-1}z^{-(jh_k+\\sum_{i=1}^ja_i^{(k)})}\\right),\\nonumber \\\\\n\\nonumber\n\\end{eqnarray*}\n\\noindent{}$\\lambda$ denotes the normalized Lebesgue measure on the\ncircle group ${\\sym T}$ and $W^{*} \\lim$ denotes weak*limit in the space of\nbounded Borel measures on ${{\\sym T}}$.\\\\ \n\n\n\\noindent{}The principal result of this paper is the following:\\\\\n\n\n\\noindent {\\bf Theorem 2.1. }{\\it Let $T=T_{(p_k,\n(a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty}$ be a rank one\ntransformation such that,\n\n\\begin{eqnarray}{\\label{eq:eqt}}\n&&(i)~~a_{j+1}^{(k)}\\geq 2s_k(j),j=0,\\cdots, p_k-1,\\nonumber\\\\\n&& {\\rm {~~with~~~~}} s_k(j)=a_1^{(k)}+\\ldots +a_j^{(k)},\ns_k(0)=0,\\nonumber\\\\\n&&(ii)~~\\frac{s_k{(p_k)}}{h_k}<\\frac12 \\nonumber\n\\end{eqnarray} then the spectrum of $T$ is singular.}\n\\\\\n\n\\noindent{}We remark that the\nspectrum of rank one transformation is always singular if the\ncutting parameter $p_k$ is bounded. In fact, Klemes-Reinhold\nprove that if $\\displaystyle \\sum_{k=0}^{\\infty}\n\\frac{1}{{p_k}^2}=\\infty$ then the associated rank one\ntransformation has singular spectrum. Henceforth, we assume that the series\n$\\displaystyle \\sum_{k=0}^{\\infty} \\frac{1}{{p_k}^2}$\nconverges.\\\\\nWe point out also that the condition $(ii)$ of\ntheorem 2.1 holds in the case of rank one transformations satisfying a\nrestricted growth condition provided that $\\displaystyle \\min_{1 \\leq j \\leq p_k}(a_j^{(k)})=0$. Following Creutz-Silva \\cite{Creutz-silva},\nwe say that a rank one transformation $T=T_{(p_k, (a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty}$ has\nrestricted growth if\n\\[\n\\displaystyle \\frac{\\displaystyle \\sum_{i=1}^{p_k} \\left( a_j^{(k)}- \\displaystyle \\min_{1 \\leq j \\leq p_k}(a_j^{(k)})\\right) }{h_k+\\displaystyle \\min_{1 \\leq j \\leq p_k}(a_j^{(k)}) }\\tend{k}{+\\infty}0,\n\\]\n\\noindent{}The proof of our main result is based on the method of\nJ. Bourgain in \\cite{Bourgain}. We shall recall the main ideas of\nthis\nmethod.\\\\\n\n\n\\noindent {\\bf Proposition 2.2. }{\\it \\ The following are\nequivalent }\n\n\\begin{enumerate}\n\\item [(i)] {\\it $\\sigma \\perp \\lambda$ }\n\n\\item[(ii)] {\\it $\\inf \\{\\displaystyle \\displaystyle \\int \\displaystyle \\int\n\\prod_{l=1}^k\\left| {P_{n_l}(z)}\\right| d\\lambda,~k\\in\n{{\\sym N}},~n_1n_k$\nand put\n\\[\nQ\\left(z\\right) =\\prod_{i=1}^k\\left |{P_{n_i}(z)}\\right |.\n\\]\n\n\\noindent{}One can show, using the same arguments as in\n\\cite{elabdal1}, the following lemma.\\\\\n\n\\noindent {\\bf Lemma 2.3. }{\\it\n\\begin{eqnarray*}\n &&\\displaystyle \\int Q(z) \\left| P_m(z)\\right|d\\lambda(z)\n\\leq \\\\\n &&\\frac 12\\left( \\displaystyle \\int Q d\\lambda +\\displaystyle \\int\nQ(z) \\left| P_m(z)\\right| ^2 d\\lambda(z)\\right) -\\frac 18\\left(\n\\displaystyle \\int Q \\left| \\left| P_m(z)\\right| ^2-1\\right|\nd\\lambda(z)\\right) ^2. \\end{eqnarray*}}\\\\\n\n\\noindent {\\bf Proposition 2.4. }{\\it $\\displaystyle \\lim_{m\n\\rightarrow\n \\infty}\\displaystyle \\int Q \\left|\nP_m(z)\\right|^2d\\lambda(z) {\\Bbb {=}} \\displaystyle \\displaystyle\n\\int Q $ $d\\lambda(z) .$}\\\\\n\\\\\n\\begin{proof}\n\\noindent{}The\nsequence of probability measures $\\left| {P_m(z)}%\n\\right|^2d\\lambda(z)$ converges weakly to the Lebsegue measure.\n \\end{proof}\n \\\\\n\n\n\\noindent We have also the following proposition: \\\\\n\n\\noindent {\\bf Proposition 2.5. }{\\it There exist a subsequence of\nthe sequence $\\left (\\left |\\left| P_m(z)\\right|-1 \\right|\\right)$\nwhich converge weakly to some non-negative function $\\phi$ which\nsatisfies $ \\phi \\leq $2, almost surely with respect to the Lebesgue\nmeasure.}\n\\\\\n\n\\begin{proof}\n\\noindent{}The\nsequence $\\left |\\left| {P_m(z)}%\n\\right|-1 \\right |$ is bounded in $L^2$. It follows that there\nexist a subsequence which converges weakly to some non-negative\n$L^2$ function $\\phi$. Let $\\omega$ be a non-negative continuous\nfunction, then we have\n\n\\begin{eqnarray*}\n\\int \\omega\\left |\\left| {P_m(z)} \\right|-1 \\right |d\\lambda(z)\n&&\\leq \\int \\omega\\left| {P_m(z)} \\right| d\\lambda(z)+\\int \\omega\nd\\lambda(z)\\\\\n&&\\leq {(\\int \\omega d\\lambda(z))}^{\\frac12} {(\\int \\omega \\left |\n{P_m(z)} \\right|^2d\\lambda(z))}^{\\frac12}+\\int \\omega d\\lambda(z).\n\\end{eqnarray*}\n\\noindent{}Hence\n\\[\n\\int \\omega \\phi d\\lambda \\leq 2 \\int \\omega d\\lambda.\n\\]\n\nSince this holds for all non-negative continuous $\\omega$, we have $\\phi \\leq 2$ a.e. \n\\end{proof}\n\\\\\n\n\\noindent Put\n\\[\n\\alpha = \\phi~~ d\\lambda.\n\\]\n\\noindent We shall prove the following:\\\\\n\n\\noindent {\\bf Proposition 2.6. }{$\\alpha \\bot \\sigma$}.\\\\\n\n\\noindent For the proof of the proposition 2.6 we need the\nfollowing classical lemma \\cite{Kilmer-Seaki}.\\\\\n\n\\noindent {\\bf Lemma 2.7. }{\\it Let $\\rho, \\tau$ be two\nnonnegative finite measure on a measurable space $X$. Then the\nfollowing properties are equivalent :\n\\begin{enumerate}\n\\item[(i)] {\\it $\\rho \\perp \\tau$ }\n\n\\item[(ii)] {\\it Given $\\varepsilon>0$, there exists a\nnonnegative measurable function $f$ on $X$ such that $ f >0,$\n$\\tau-a.e.$ and such that\n\\[\n\\left(\\int f d\\rho) \\right) \\left (\\int \\frac{d\\tau}{f} \\right)\n<\\varepsilon.\n\\] }\n\\end{enumerate}\n}\n\n\\begin{proof}{of Proposition 2.6}\nLet $\\displaystyle \\beta_1=\\inf\\{\\int Q\nd\\alpha,~:~Q=\\prod_{i=1}^k\\left | {P_{n_i}(z)}\\right|, k \\in {\\sym N},\nn_1n_k$.\nThen, by Cauchy-Schwarz inequality, we have\n\n\\begin{eqnarray}{\\label{eq:Bo1}}\n\\int \\prod_{j=1}^{N}|P_j|d\\alpha & =& \\int \\sqrt{\\prod_{j \\in\n\\mathcal{N}}|P_j|} \\sqrt{\\prod_{j \\in \\mathcal{N}}|P_j|} \\prod_{j\n\\not \\in \\mathcal{N}}|P_j| d\\alpha \\nonumber\\\\\n&\\leq& {\\left(\\int \\prod_{j \\in \\mathcal{N}}|P_j|d\\alpha\\right\n)}^{\\frac12} {\\left(\\int \\prod_{j \\in \\mathcal{N}} |P_j| d\\alpha\n\\prod_{j \\not \\in \\mathcal{N}}|P_j|^2 d\\alpha\\right )}^{\\frac12}\n\\end{eqnarray}\n\n\\noindent{}But\n\n\\begin{eqnarray}{\\label{eq:B2}} \\int \\prod_{j \\in \\mathcal{N}}\n|P_j| \\prod_{j \\not \\in \\mathcal{N}}|P_j|^2 d\\alpha &\\leq & 2\n\\int \\prod_{j \\in \\mathcal{N}} |P_j| \\prod_{j \\not \\in\n\\mathcal{N}}|P_j|^2 d\\lambda \\nonumber \\\\\n&\\leq& 2\\int \\prod_{j=1}^{N} |P_j| \\prod_{j \\not \\in\n\\mathcal{N}}|P_j| d\\lambda \\nonumber\\\\\n &\\leq& 2{\\left(\\int \\prod_{j=1}^{N} |P_j|^2 d\\lambda \\right)}^{\\frac12}\n{\\left(\\int\\prod_{j \\not \\in \\mathcal{N}}|P_j|^2\nd\\lambda\\right)}^{\\frac12} \\nonumber \\\\\n&\\leq& 2\n\\end{eqnarray}\n\n\\noindent{}Combine (\\ref{eq:Bo1}) and (\\ref{eq:B2}) to get the\nclaim. The proof of the proposition follows from lemma 2.7.\n\\end{proof}\n\n\n\\section*{3. Estimation of $L_1$-Norm of ($\\left |P_m(z)\\right |-1$)\nand the Central Limit Theorem.}\n\nIn this section we assume that for $m$\nsufficiently large\n\\begin{eqnarray}{\\label{eq:rg}}\n &&(i)~~a_{j+1}^{(m)}\\geq 2s_m(j),j=0,\\cdots, p_m-1,\\nonumber\n \\\\ &&{\\rm {~~with~~~~}} s_m(j)=a_1^{(m)}+\\ldots +a_j^{(m)},\ns_m(0)=0,\\nonumber\\\\\n&&(ii)~~\\frac{s_m{(p_m)}}{h_m}<\\frac12 \\nonumber\n\\end{eqnarray}\n\n\\noindent{}Under the above assumptions, we shall proof the following:\\\\\n\n\\noindent {\\bf Proposition 3.1. }{ $\\alpha \\geq K \\lambda$, for\nsome positive\nconstant $K$}.\\\\\n\n\\noindent{} The proof of the proposition is based on an\nestimate of $\\displaystyle \\int_A ||P_m|-1|d\\lambda$, where\n$A$ is a Borel set with $\\lambda(A)>0$. More precisely we shall\nstudy the stochastic behavior of the sequence $ |P_m|$. For that\nour principal strategy is based on the method of the central\nlimit theorem for trigonometric sums. A nice account can be\nfound in \\cite{Kac}. It is well-known that\nHadamard lacunary trigonometric seires satisfies the central limit\ntheorem \\cite{Zygmund}. The central limit theorem for\ntrigonometric sums has been studied by many authors, Zygmund and\nSalem \\cite{Zygmund}, Erd\\\"os \\cite{Erdos},\n J.P. Kahane \\cite{Kahane}, Berkers \\cite{Berkes}, Murai \\cite{Murai},\nTakahashi \\cite{Takahashi}, Fukuyama and Takahashi\n\\cite{Fukuyama}, and many others. The same method is used to\nstudy the asymptotic behavior of Riesz-Raikov sums \\cite{petit}.\\\\\n\n\\noindent Here, we shall prove the following:\\\\\n\n\\noindent {\\bf Proposition 3.2. } {If $(i)$ and $(ii)$ hold then\nfor any Borel subset $A$ of ${\\sym T}$ with $\\lambda(A)>0$, the distribution of the\nsequence of random variables\n$\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\sum_{j=0}^{p_m-1} \\cos((jh_m+s_m(j))t)$\nconverges to the Gauss distribution. That is\n\n\\begin{eqnarray}{\\label{eq:eq6}}\n\\frac1{\\lambda(A)}\\left \\{t \\in A~~:~~\\frac{\\sqrt{2}}{\\sqrt{p_m}}\n\\sum_{j=0}^{p_m-1} \\cos((jh_m+s_m(j))t) \\leq x \\right \\}\\nonumber \\\\\n\\tend{m}{\\infty}\\frac1{\\sqrt{2\\pi}}\\int_{-\\infty}^{x}\ne^{-\\frac12t^2}dt\\stackrel{\\rm {def}}{=}{\\mathcal {N}}\\left( \\left] -\\infty,x \\right] \\right) .\n\\end{eqnarray}\n}\\\\\nThe proof of proposition 3.2 is based on the idea of the proof of\nmartingale central limit theorem due to McLeish \\cite{Mcleish}.\nThe main ingredient is the following .\\\\\n\n\\noindent {\\bf Lemma 3.3. } {For $n \\geq 1$, let $U_n,T_n$ be\nrandom variables such that\n\\begin{enumerate}\n\\item $U_n\\longrightarrow a$ in probability.\n\\item $\\{T_n\\}$ is uniformly integrable.\n\\item $\\{|T_nU_n|\\}$ is uniformly integrable.\n\\item ${\\sym E}(T_n) \\longrightarrow 1$.\n\\end{enumerate}\nThen ${\\sym E}(T_nU_n)\\longrightarrow a$.}\\\\\n\n\\begin{proof}\nWrite $T_nU_n=T_n(U_n-a)+aT_n$. As ${\\sym E}(T_n) \\longrightarrow\n1$, we need to show that ${\\sym E}(T_n(U_n-a))\\longrightarrow 0$.\\\\\nSince $\\{T_n\\}$ is uniformly integrable, we have\n$T_n(U_n-a)\\longrightarrow 0$ in probability. Also, both $T_nU_n$\nand $aT_n$ are uniformly integrable, and so the combination\n$T_n(U_n-a)$ is uniformly integrable. Hence,\n${\\sym E}(T_n(U_n-a))\\longrightarrow 0$.\n\\end{proof}\\\\\n\n\\noindent{}Let us recall the following expansion\n\\[\n\\exp(ix)=(1+ix)\\exp(-\\frac{x^2}2+r(x)),\n\\]\n\\noindent{}where $|r(x)| \\leq |x|^3$, for real $x$.\\\\\n\n\\noindent {\\bf Theorem 3.4. }{Let $\\{X_{nj}~~:~~ 1 \\leq j \\leq\nk_n, n\\geq 1\\}$ be a triangular array of random variables.\n$\\displaystyle S_n=\\sum_{j=1}^{k_n}X_{nj}$, $\\displaystyle T_n=\\prod_{j=1}^{k_n}(1+itX_{nj})$,\nand $\\displaystyle U_n=\\exp \\left (-\\frac{t^2}2 \\sum_j\nX_{nj}^2+\\sum_{j}r(tX_{nj})\\right )$. Suppose that\n\\begin{enumerate}\n\\item $\\{T_n\\}$ is uniformly integrable.\n\\item ${\\sym E}(T_n) \\longrightarrow 1$.\n\\item $\\displaystyle \\sum_j X_{nj}^2 \\longrightarrow 1$ in probability.\n\\item $\\max |X_{nj}|\\longrightarrow 0$ in probability.\n\\end{enumerate}\nThen ${\\sym E}(\\exp(it S_n))\\longrightarrow \\exp(-\\displaystyle\\frac{t^2}2).$}\\\\\n\n\\begin{proof}\nLet $t$ be fixed. From condition $(3)$ and $(4)$,\n\\[\n|\\sum_j r(tX_{nj})| \\leq |t|^3 \\sum_{j}|X_{nj}|^3 \\\\\n\\leq |t|^3 \\max_j|X_{nj}| \\sum_j X_{nj}^2\n\\tend{n}{\\infty}0 {\\rm {~in~probability}}.\n\\]\n\\noindent{}$U_n=\\exp\\left (-\\displaystyle \\frac{t^2}2 \\sum_j\nX_{nj}^2+\\sum_{j}r(tX_{nj})\\right ) \\longrightarrow \\exp(-\\displaystyle\n\\frac{t^2}2)$ in probability as $n \\longrightarrow +\\infty$. This verifies the condition $(1)$ of Lemma\n3.3 with $a=\\exp(-\\displaystyle \\frac{t^2}2).$ It is easy to check\nthat the conditions $(2)$ ,$(3)$ and $(4)$ of the lemma 3.3 hold.\nThus $E\\left( \\exp(itS_n)\\right) =E\\left( T_nU_n\\right) \\longrightarrow\n\\exp(-\\displaystyle \\frac{t^2}2)$.\n\\end{proof}\\\\\n\n\\noindent{}Let $m$ be a positive integer and put\n\n\\begin{eqnarray*}\nW_m \\stackrel {\\rm {def}}{=}&& \\big \\{ \\big ({\\sum_{i \\in\nI}\\varepsilon_i p_i}\\big ) h_m+\\sum_{i \\in I}\\varepsilon_i\ns_m(p_i) ~~:~~\n\\varepsilon_i \\in \\{0,-1,1\\}, I \\subset \\{0,\\cdots,p_m-1\\},\\\\\n&&p_i \\in \\{0,\\cdots,p_m-1\\} \\}.\n\\end{eqnarray*}\n\n\\noindent{} The element $w =(\\sum_{i \\in\nI}\\varepsilon_ip_i)h_m+\\sum_{i \\in I}\\varepsilon_is_m(p_i)$ is\ncalled a word.\\\\\n\nWe shall need the following combinatorial lemma.\\\\\n\n\\noindent {\\bf Lemma 3.5.}{ Under the\nassumptions (i) and (ii) of theorem 2.1, all the words of $W_m$ are distinct.}\\\\\n\n\\begin{proof}{}\nLet $w,w' \\in W_m$, write\n\\begin{eqnarray*}\nw &=&(\\sum_{i \\in I}\\varepsilon_ip_i)h_m+\\sum_{i \\in\nI}\\varepsilon_is_m(p_i)\\\\\nw' &=&(\\sum_{i \\in I'}\\varepsilon'_ip'_i)h_m+\\sum_{i \\in\nI}\\varepsilon'_is_m(p'_i).\n\\end{eqnarray*}\n\\noindent{}Then $w=w'$ implies\n\\[\n\\{(\\sum_{i \\in I}\\varepsilon_ip_i)-(\\sum_{i \\in\nI'}\\varepsilon'_ip'_i)\\}h_m=\\sum_{i \\in\nI'}\\varepsilon'_is_m(p'_i)-\\sum_{i \\in I}\\varepsilon_is_m(p_i)\n\\]\n\\noindent{}But the modulus of LHS is greater than $h_m$ and the\nmodulus of RHS is less than $2 \\sum_{j=0}^{p_m-1}a_j^{(m)}$. It\nfollows that\n\\begin{eqnarray*}\n\\sum_{i \\in I}\\varepsilon_ip_i&=& (\\sum_{i \\in\nI'}\\varepsilon'_ip'_i){\\rm {~~~and }} \\\\\\sum_{i \\in\nI}\\varepsilon_is_m(p_i)&=&\\sum_{i \\in I'}\\varepsilon'_is_m(p'_i).\n\\end{eqnarray*}\nSince from (i) we have $s_m(p+1) \\geq 3 s_m(p)$, we get that the\nrepresentation in the form $\\sum_{i \\in I}\\varepsilon_is_m(p_i)$\nis unique and the proof of the lemma is complete.\n\\end{proof}\n\\\\\n\n\\begin{proof}{of proposition 3.2} Let $A$ a Borel set,\n$\\lambda(A)>0$. Using the Helly theorem we may assume that the\nsequence $\\displaystyle \\frac{\\sqrt{2}}{\\sqrt{p_m}}\n\\sum_{j=0}^{p_m-1} \\cos((jh_m+s_m(j))t)$ converge in distribution.\nAs is well-known, it is sufficient to show that for every real\nnumber $x$,\n\n\\begin{eqnarray}{\\label {eq:eq7}}\n \\displaystyle \\frac1{\\lambda(A)}\\int_A \\exp\\left \\{-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\n \\sum_{j=0}^{p_m-1} \\cos((jh_m+s_m(j))t) \\right \\} dt\n \\tend{m}{\\infty} \\exp(-\\frac{x^2}2). \\nonumber\n\\end{eqnarray}\n\n\\noindent{} To this end we apply theorem 3.4. in the following\ncontext. The measure space is the given Borel set $A$ of positive Lebesgue\nmeasure in the circle with the normalised measure and the random\nvariables are given by\n$$\nX_{mj}=\\frac{\\sqrt{2}}{\\sqrt{p_m}} \\cos((jh_m+s_m(j))t),~~~~{\\rm {where}}~~~0 \\leq j \\leq p_m-1,~\nm \\in {\\sym N}.\n$$\n\n\\noindent{}It is easy to check that the variables $\\{X_{mj}\\}$\nsatisfy conditions (1) and (4). Further, condition (3) follows from fact that\n$$\n\\int_0^{2\\pi} \\left |\\sum_{j=0}^{p_m-1} X_{mj}^2-1 \\right |^2 dt\n\\tend{m}{\\infty} 0.\n$$\n\\noindent{} It remains to verify comdition (2) of theorem 3.4. it is sufficient to show that\n\\begin{eqnarray}{\\label {eq8}}\n\\int_A \\prod_{j=0}^{p_m-1}\\left (\n1-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\cos((jh_m+s_m(j))t)\\right)dt\n\\tend{m}{\\infty}\\lambda(A).\n\\end{eqnarray}\n\n\n\\noindent{}Write\n\\begin{eqnarray*}\n\\Theta_m(x,t)&=&\\prod_{j=0}^{p_m-1}\n\\left(1-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\cos((jh_m+s_m(j))t\\right)\\\\\n&=&1+\\sum_{w=1}^{N_m}{\\rho_w}^{(m)}(x) \\cos(wt),\n\\end{eqnarray*}\n\n\\noindent{}where $\\rho_w=0$ if $w$ is not of the form $(\\sum_{i\n\\in I}\\varepsilon_ip_i)h_m+\\sum_{i \\in I}\\varepsilon_is_m(p_i)$,\n$N_m=\\displaystyle \\left \\{\\frac{p_m(p_m-1)}2\\right\n\\}h_m+s_m(p_m-1)+s_m(p_m-2)+\\cdots+1$.\\\\\n\n\\noindent{}We claim that it is sufficient to prove the following:\n\n\\begin{eqnarray}{\\label{density}}\n\\int_{0}^{2\\pi} R(t) \\prod_{j=0}^{p_m-1} \\left\n(1-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\cos((jh_m+s_m(j))t)\\right) dt\n\\tend{m}{\\infty}\\int_{0}^{2\\pi} R(t) dt,\n\\end{eqnarray}\n\n\\noindent{}for any trigonometric polynomial $R$. In fact, assume\nthat (\\ref{density}) holds and let $\\epsilon>0$.\nThen, by the density of trigonometric polynomials, one can find a\ntrigonometric polynomial $R_\\epsilon$ such that\n\\[\n||\\chi_A-R_\\epsilon||_1 <\\epsilon,\n\\] where\n\\noindent{}$\\chi_A$ is indicator function of $A$. But\n\n\\begin{eqnarray*}\n\\left |\\prod_{j=0}^{p_m-1}\\left (\n1-ix\\frac{\\sqrt{2}}{\\sqrt{p_m}}\\cos((jh_m+s_m(j))t)\\right)\\right| \\leq\n\\left \\{\\prod_{j=0}^{p_m-1}\\left ( 1+\\frac{2x^2}{p_m}\\right) \\right\n\\}^{\\frac12},\n\\end{eqnarray*}\n\n\\noindent{}Since $1+u \\leq e^u$, we get\n\n\\begin{eqnarray}{\\label{eq:eq11}}\n\\mid\\Theta_m(x,t)\\mid \\leq e^{x^2}.\n\\end{eqnarray}\n\n\\noindent{}Hence, according to (\\ref{density}), for $m$\nsufficiently large, we have\n\n\\begin{eqnarray*}{\\label{eq :eq12}}\n&&\\left |\\int_A \\Theta_m(x,t) dt -\\lambda(A)\\right|\n=|\\int_A\\Theta_m(x,t)dt -\\int_{0}^{2\\pi} \\Theta_m(x,t) R_\\epsilon(t) dt+\n\\\\&&\\int_{0}^{2\\pi} \\Theta_m(x,t) R_\\epsilon(t) dt-\\int_{0}^{2\\pi} R_\\epsilon(t) dt + \\int_{0}^{2\\pi}\nR_\\epsilon(t) dt-\\lambda(A)|< e^{x^2}\\epsilon+2\\epsilon.\n\\end{eqnarray*}\n\n\\noindent{}The proof of the claim is complete. It still remains to prove\n(\\ref{density}). Observe that\n\n$$\\int_{0}^{2\\pi} \\Theta_m(x,t)R(t) dt =\n \\int_{0}^{2\\pi} R(t) dt+\\sum_{w=1}^{N_m}{\\rho_w}^{(m)}(x) \\int_{0}^{2\\pi} R(t)\n \\cos(wt)dt$$\n\n\\noindent{}and for\n$w=p_{i_1}h_m+s_m(p_{i_1})+\\sum_{j=1}^{r}\\varepsilon_j\n\\{(p_{i_j}h_m)+s_m(p_{i_j})\\}$, we have\n\n\\[\n|{\\rho_w}^{(m)}(x)| \\leq \\frac{2^{1-r}|x|^r}{p_m^{\\frac{r}2}},\n\\]\n\n\\noindent{}hence\n\\[\n\\max_{w \\in W}|{\\rho_w}^{(m)}(x)| \\leq\n\\frac{|x|}{p_m^{\\frac{1}2}}\\tend{m}{\\infty}0.\n\\]\n\\noindent{}Since $\\displaystyle \\sum_{w \\in{\\sym Z}}|\\int_{0}^{2\\pi} e^{-iwt}\nR(t)dt|$ is bounded, we deduce\n\\[\n|\\sum_{w=1}^{N_m}{\\rho_w}^{(m)}(x) \\int_{0}^{2\\pi} R(t)\n \\cos(wt)dt|\\leq \\frac{|x|}{p_m^{\\frac{1}2}}\n \\sum_{w \\in{\\sym Z}}|\\int_{0}^{2\\pi} e^{-iwt} R(t)dt|\\tend{m}{\\infty}0.\n\\]\n\\noindent{}The proof of the proposition 3.2. is complete.\n\\end{proof}\\\\\n\n\\begin{proof}{of proposition 3.1}\nLet $A$ be a Borel subset of ${\\sym T}$, and $x \\in ]1,+\\infty[$, then, for any\npositive integer $m$, we have\n\n\\begin{eqnarray*}{\\label{eq :eqfin}}\n\\int_A ||P_m(\\theta)|-1| d\\lambda(\\theta) &\\geq& \\int_{\\{\\theta\n\\in A\n~~~:~~~~|P_m(\\theta)|>x\\}}||P_m|-1| d\\lambda(\\theta)\\\\\n&\\geq&(x-1)\\lambda\\{\\theta \\in A~:~|P_m(\\theta)|>x\\}\\\\\n&\\geq& (x-1) \\lambda\\{\\theta \\in A ~~:~~|\\Re({P_m(\\theta)})|>x\\}\n\\end{eqnarray*}\n\\noindent{} Let $m$ go to infinity and use propositions\n2.5 and 3.2 to get\n\n\\[\n\\int_A \\phi d\\lambda \\geq (x-1)\\{1-{\\mathcal\n{N}}([-\\sqrt{2}x,\\sqrt{2}x])\\}\\lambda(A).\n\\]\n\n\\noindent{}Put $K=(x-1)\\{1-{\\mathcal {N}}([-\\sqrt{2}x,\\sqrt{2}x])\\}$. Hence $\n\\alpha(A) \\geq K \\lambda(A)$, for any Borel subset $A$ of ${\\sym T}$ which proves the proposition.\n\\end{proof}\\\\\n\n\\noindent{}Now, We give the proof of our main result.\\\\\n\n\\begin{proof}{of theorem 2.1} Follows easily from the proposition 2.6 combined with\nproposition 3.1. \n\\end{proof}\\\\\n\n\\noindent{}Let us mention that the same proof works for the\nfollowing more general statement.\\\\\n\n\\noindent {\\bf Theorem 3.6. }{\\it Let $T=T_{(p_k,\n(a_j^{(k)})_{j=1}^{p_k})_{k=0}^\\infty}$ be a rank one\ntransformation such that,\n\n\\begin{eqnarray}{\\label{eq:eqt2}}\n&&(i)~~a_{j+1}^{(k)}\\geq 2s_k(j),j=0,\\cdots, p_k-1,\\nonumber\\\\\n&& {\\rm {~~with~~~~}} s_k(j)=a_1^{(k)}+\\ldots +a_j^{(k)},\ns_k(0)=0,\\nonumber\\\\\n&&(ii)~~\\frac{s_k{(p_k)}-p_k\\min_{1 \\leq j \\leq\np_k}(a_j^{(k)})}{h_k+\\min_{1 \\leq j \\leq p_k}(a_j^{(k)})}<\\frac12\n\\nonumber\n\\end{eqnarray} then the spectrum of $T$ is singular.}\n\\\\\n\n\n\\noindent{}{\\bf Remark.} We note that, in \\cite{Bourgain} and\n\\cite{Klemes1}, the strategy of the authors is to show that the\nabsolutely continuous measure $\\beta$, obtained as the limit of\nsome subsequence of the sequence $(||P_m|^2-1| d\\lambda)_{m \\geq\n0}$, is equivalent to Lebesgue measure, in fact the authors prove\nthat\n$$ \\beta \\geq K \\lambda,~~~~~~~~~~~~~~~~ (E)$$\n\\noindent{}for some $K>0$. In the case of Ornstein\ntransformations, the relation (E) hold almost surely.\\\\\n\n\n\n\\section*{4. Simple Proof Of Bourgain Theorem.}\n\nBourgain Theorem deals with Ornstein transformations. In\nOrnstein's construction, the $p_k$'s are rapidly increasing, and\nthe number of spacers, $a_i^{(k)}$, $1 \\leq i\\leq p_k-1$, are\nchosen randomly. This may be organized in different ways as\npointed out by J. Bourgain in \\cite{Bourgain}. Here we suppose that we are given\ntwo sequences $(t_k)$, $(p_k)$ of positive integers and a sequence ($\\xi_k$) \n of probability measure such that the support of each\n$\\xi_k$ is a subset of $X_k = \\{-\\displaystyle\n\\frac{t_k}{2},\\cdots,\\displaystyle \\frac{t_k}{2}\\}$. We choose now\nindependently, according to $\\xi_k$, the numbers\n$(x_{k,i})_{i=1}^{p_k-1}$, and $x_{k,p_k}$ is chosen\ndeterministically in ${\\sym N}$. We put, for $1 \\leq i \\leq p_k$,\n\n$$a_i^{(k)} = t_{k} + x_{k,i} - x_{k,i-1}, ~~{\\rm with} ~~x_{k,0}\n= 0.$$\\\\\n\n\\noindent{} We have\n\n$$h_{k+1} = p_k(h_k + t_{k}) + x_{k,p_k}.$$\n\n\\noindent{}So the deterministic sequences of positive integers\n$(p_k)_{k=0}^\\infty$, $(t_k)_{k=0}^\\infty $ and\n$(x_{k,p_k})_{k=0}^\\infty$ determine completely the sequence of\nheights $(h_k)_{k=0}^\\infty$. The total measure of the resulting\nmeasure space is finite if\n\\begin{eqnarray}\\label{eqn:fini}\n\\sum_{k=0}^{\\infty}\\frac{t_k}{h_k}+\\sum_{k=0}^\\infty\n\\frac{x_{k,p_k}}{p_kh_k} < \\infty.\n\\end{eqnarray}\n \\noindent{}We will assume that this\nrequirement is satisfied.\\\\\nWe thus have a probability space of Ornstein transformations\n$\\Omega=\\prod_{l=0}^\\infty X_l^{p_l-1}$ equipped with the natural\nprobability measure ${\\sym P} \\stackrel {\\rm def}\n{=}\\otimes_{l=1}^{\\infty} P_l$, where $P_l\\stackrel {\\rm def}\n{=}\\otimes_{j=1}^{p_l-1}{\\xi_j}$; ${\\xi_j}$ is the probability\nmeasure on $X_j$. We denote this space by $(\\Omega, {\\mathcal\n{A}}, {{\\sym P}})$. So $x_{k,i}$, $1 \\leq i \\leq p_k -1$, is the\nprojection from $\\Omega$ onto the $i^{th}$ co-ordinate space of\n$\\Omega_k \\stackrel {\\rm def} {=} X_k^{p_k-1}$, $1 \\leq i \\leq\np_k-1$. Naturally each point $\\omega =(\\omega_k =\n(x_{k,i}(\\omega))_{i=1}^{p_k-1})_{k=0}^\\infty$ in $\\Omega$ defines\nthe spacers and therefore a rank one transformation\n$T_{\\omega,x}$, where $x=(x_{k,p_k})$.\n\n\\noindent{}This construction is more general than the\n construction due to Ornstein \\cite{Ornstein} which corresponds to the case \n$t_k=h_{k-1}$, $\\xi_k$ is the uniform distribution on $X_k$ and $p_k >> h_{k-1}$. \\\\\n\n\n\\noindent{} We recall that Ornstein in \\cite{Ornstein} proved that\nthere exists a sequence ${(p_k,x_{k,p_k})}_{k \\in {\\sym N}}$ such that,\n$T_{\\omega,x}$ is almost surely mixing. Later in \\cite{Prikhodko}\nPrikhod'ko obtains the same result for some special choice of the\nsequence of the distribution ${(\\xi_m)}$ and recently, using the\nidea of D. Creutz and C. E. Silva \\cite{Creutz-silva} one can\nextend this result to a large family of probability measures\nassociated to Ornstein construction. In our general construction,\naccording to (\\ref{eqn:type1}) the spectral type of each\n$T_{\\omega}$ , up to a discrete measure, is given by\n \\[\n\\sigma_{T_\\omega }=\\sigma^{(\\omega)}\n_{\\chi_{B_0}}=\\sigma^{(\\omega)} =W ^{*}\\lim \\prod_{l=1}^N\\frac\n1{p_l}\\left| \\sum_{p=0}^{p_l-1}z^{p(h_l+t_l)+x_{l,p}}\\right|\n^2d\\lambda.\n\\]\n\n\\noindent{} With the above notations, we state Bourgain theorem in\nthe following\nform:\\\\\n\n\\bigskip\n\\noindent {\\bf Theorem 4.1 (\\cite{elabdal2}). }{\\it For every\nchoice of $(p_k), (t_k), (x_{k,p_k})$ and for any family of\nprobability measures ${\\xi_k}$ on $X_k=\\{-t_k,\\cdots,t_k\\}$ of\n${\\sym Z}$, $ {k \\in {\\sym N}^*}$, for which\n$$\\sum_{s \\in X_m}\\xi(s)^2 \\tend{m}{\\infty}0,$$ \\noindent{}the\nassociated generalized Ornstein transformations has almost surely\nsingular spectrum. i.e.,\n\\[\n{\\sym P}\\{\\omega~:~ \\sigma^{(\\omega)} ~\\bot ~\\lambda\\}=1.\n\\]\n\\noindent{}where ${\\sym P} \\stackrel {\\rm def} {=}\\otimes_{l=0}^{\\infty}\n\\otimes_{j=1}^{p_l-1}{\\xi_l}$ is the probability measure on\n$\\Omega=\\prod_{l=0}^\\infty X_l^{p_l-1}$. } \\vskip0.5cm\n\n\n\\noindent{}In the context of Ornstein's constrution, we state\nproposition 2.5 in the following form:\\\\\n\n\\noindent {\\bf Proposition 4.2. }{\\it There exist a subsequence of\nthe sequence $\\left (\\left |\\left| P_m(z)\\right|-1 \\right|\\right)$\nwhich converge weakly to some non-negative function\n$\\phi(\\omega,\\theta)$ which satisfies $ \\phi \\leq $2, almost surely\nwith respect the ${\\sym P} \\bigotimes \\lambda$.}\n\\\\\n\n\\begin{proof}\nEasy exercise.\n\\end{proof}\\\\\n\n\n\\noindent Put, for any $\\omega\\in \\Omega$\n\\[\n\\alpha_{\\omega}=\\phi(\\omega,\\theta)d\\lambda.\n\\]\nWe shall prove that $\\alpha_{\\omega}$ is equivalent to the\nLebesgue measure for almost all $\\omega$. In fact, we have the\nfollowing proposition:\n\\\\\n\n\\noindent {\\bf Proposition 4.3. }{\\it There exist an absolutely\npositive constant $K$ such that\n for almost all $\\omega$ we have\n$$\\alpha_{\\omega} \\geq K \\lambda.$$}\n\\\\\n\n\\noindent The proof is based on the following two lemmas.\n\n\\noindent {\\bf Lemma 4.4. }{\\it $\\displaystyle lim \\int \\int\n||P_m|-|P_m'|| d\\theta d{\\sym P} \\tend{n} {\\infty} 0$, where $\\displaystyle\nP_m'(\\theta)=P_m(\\theta)-\\int_{\\Omega}P_m(\\theta)d{\\sym P}.$ }\n\n\n\\begin{proof} For any $m \\in {\\sym N}$, we have\n\\begin{eqnarray*}\n\\int \\int ||P_m|-|P_m'|| d\\theta d{\\sym P} &\\leq& \\int \\int |P_m-P_m'|\nd\\theta d{\\sym P} \\\\ & = &\n\\int |\\int P_m d{\\sym P}| d\\theta\\\\\n&\\leq& \\int |\\sum_{p=0}^{p_m-1}\\frac1{\\sqrt{p_m}} z^{p(h_m+t_m))}| |\\sum_{s\\in\nX_{m}}\\xi_m(s)z^s| dz.\n\\end{eqnarray*}\nHence by Cauchy-Schwarz inequality\n\\begin{eqnarray*}\n\\int \\int ||P_m|-|P_m'|| d\\theta d{\\sym P} \\leq \\sum_{s \\in X_m}\\xi(s)^2\n\\tend{m}{\\infty}0.\n\\end{eqnarray*}\nThe proof of the lemma is complete.\n\\end{proof}\\\\\n\\vskip0.5cm \\noindent Now observe that we have\n\\[\n\\int_T |\\sum_{s\\in X_{m}}\\xi_m(s)z^s|^2 dz=\n\\int_T |\\sum_{s\\in X_{m}}\\xi_m(s)z^{2s}|^2 dz=\\sum_{s \\in X_m}\\xi(s)^2\n\\tend{m}{\\infty}0.\n\\]\nSo, we may extract a subsequence $(m_k)$ for which, for almost all\n$t \\in [0,2\\pi)$, we have\n\\[\n\\sum_{s\\in X_{m_k}}\\xi_{m_k}(s)e^{ i s t} \\tend{k}{\\infty}0\n{\\rm {~~and~~}} \\sum_{s\\in X_{m_k}}\\xi_{m_k}(s)e^{ i s 2t} \\tend{k}{\\infty}0\n\\]\n\\noindent Define\n\\[\n\\Theta\n\\stackrel{def}{=}\\{\\theta ~:~ \\sum_{s\\in X_{m_k}}\\xi_{m_k}(s)e^{i\n s \\theta}\\tend{k}{\\infty}0 {\\rm {~and~}} \\sum_{s\\in X_{m_k}}\\xi_{m_k}(s)e^{ i s 2t}\n \\tend{k}{\\infty}0\\}\n\\]\n\\noindent Choose $m \\in \\{m_k\\}$, $t \\in \\Theta$ and put, for $j \\in\n\\{0,\\cdots,p_m-1\\}$,\n\\begin{eqnarray*}\nY_{m,j}(\\omega)&=&\\cos((j(h_m+t_m)+x_{m,j}(\\omega))t)-\\int\n\\cos((j(h_m+t_m)+x_{m,j}(\\omega))t) d{\\sym P},\\\\\nZ_{m,j}(\\omega)&=&\\sqrt{\\frac{2}{p_m}}Y_{m,j}(\\omega).\n\\end{eqnarray*}\n\\\\\n\\noindent {\\bf Lemma 4.5. }{\\it For any fixed $t \\in \\Theta$, the\ndistribution of the sequence of random variables $\\displaystyle\n\\sum_{j=0}^{p_m-1} Z_{m,j}(\\omega)$ converges\nto the Gauss distribution. That is,\n\\\\\n\\begin{eqnarray}{\\label{eq:beq6}}\n{\\sym P} \\left \\{\\omega \\in \\Omega~~:~~\n\\sum_{j=0}^{p_m-1} Z_{m,j}(\\omega) \\leq x\n\\right \\} \\tend{m}{\\infty}\\frac1{\\sqrt{2\\pi}}\\int_{-\\infty}^{x}\ne^{-\\frac12u^2}du\\stackrel{\\rm {def}}{=}{\\mathcal {N}}(]-\\infty,x]).\n\\nonumber\n\\end{eqnarray}\n}\n\\begin{proof}\nSince the random variables are independent, centred and uniformly bounded by $\\displaystyle\n\\frac{2\\sqrt{2}}{\\sqrt{p_m}}$, the conditions (1), (2) and (4) of the\ntheorem 3.4 are satisfied. We have also the following:\n\\begin{eqnarray*}\n&&{\\sym E} \\left \\{ \\left(\\sum_{j=1}^{p_m-1}{Z_{m,j}^2-1}\\right) ^{2} \\right\\rbrace \\\\\n&& = \\frac{4(p_m-1)}{p_m^2}{\\sym E} ( Y_{m,1}^4)+\\frac{(p_m-1)(p_m-2)}{p_m^2}\n({\\sym E} (2Y_{m,1}^2))^2-2\\frac{p_m-1}{p_m} {\\sym E} (2 Y_{m,1}^2)+1.\n\\end{eqnarray*}\n\\noindent{}But ${\\sym E}(2Y_{m,1}^2) \\tend{m}{\\infty}1$, it follows\nthat the variables $\\{Z_{m,j}\\}$ satisfy condition (3) of theorem 3.4.\nThus all the conditions of theorem 3.4 hold and we conclude that the distribution of\n$\\displaystyle \\sum_{j=0}^{p_m-1} Z_{m,j}(\\omega)$\nconverges to normal distribution.\n\\end{proof}\\\\\n\n\\begin{proof}{of proposition 4.3}\nLet $A$ be a Borel subset of ${\\sym T}$, $C$ a cylinder set in\n$\\Omega$, and $x \\in ]1,+\\infty[$, then, for any positive integer\n$m$, we have\n\n\\begin{eqnarray*}{\\label{eq :beqfin}}\n&&\\int_{A \\times C} ||P_m(\\theta)|-1| d\\lambda(\\theta) d{\\sym P} \\\\\n&\\geq& {\\sym P}(C) \\int_{A \\times \\Omega} ||P'_m(\\theta)|-1|\nd\\lambda(\\theta) d{\\sym P}\n-\\int||P_m|-|P_m'||d{\\sym P} d\\lambda\\\\\n&\\geq& {\\sym P}(C) \\int_{ \\{|P'_m|>x\\} \\bigcap A \\times\n\\Omega}||P'_m|-1| d\\lambda(\\theta)d{\\sym P}\n-\\int||P_m|-|P_m'||d{\\sym P} d\\lambda\\\\\n&\\geq&{\\sym P}(C) (x-1)\\int_A{\\sym P}\\{|\\Re\n{(P'_m(\\theta))}|>x\\}d\\lambda-\\int||P_m|-|P_m'||d{\\sym P} d\\lambda\n\\end{eqnarray*}\n\\noindent{} Let $m$ go to infinity and combine lemmas 4.4 and\n4.5 to get\n\n\\[\n\\int_{A \\times C} \\phi d\\lambda d{\\sym P} \\geq (x-1)\\{1-{\\mathcal\n{N}}([-\\sqrt{2}x,\\sqrt{2}x])\\}\\lambda(A) {\\sym P}(C).\n\\]\n\n\\noindent{}Put $K=(x-1)\\{1-{\\mathcal {N}}([-\\sqrt{2}x,\\sqrt{2}x])\\}$. Hence, for\nalmost all $\\omega$, we have, for any Borel set $A \\subset {\\sym T}$, $ \\alpha_{\\omega}(A) \\geq K\n\\lambda(A)$, and the proof of the\nproposition is complete.\n\\end{proof}\\\\\n\n\\begin{proof}{of theorem 4.1}\nFollow easily from the proposition 4.3 combined with the\nproposition 2.6.\n\\end{proof}\n\\\\\n\n\\noindent{}{\\bf Remark.} We point out that there exist a rank one \nmixing transformations on the space with finite measure satisfying\nthe condition of theorem 3.6, In fact, following the notations of section 4, one may define\nthe spacers in the Ornstein construction by\n\\[\n {a_j^{(k)}}={3^j} t_k+x_{k,j}-x_{k,j-1},\n\\]\n\n\\noindent{}and choose the sequence $(t_k)_{k \\in {\\sym N}}$ such that the measure of dynamical system is finite.\nThus the condition of theorem 3.6 hold and the class is\nmixing almost surely.\\\\\n\n\\begin{center} {\\bf Acknowledgements}\n\\end{center}\nI would like to express my thanks to J-P. Thouvenot who posed to me\nthe problem of the singularity of the spectrum of the rank one\ntransformations, and also to B. Host, F. Parreau and F. Bassam for\nmany conversations on the subject I have had with them. My thanks\nalso to Andrew Granville and the organizers of SMS-NATO ASI 2005\nSummer School.\\\\\nI am also grateful to the referee for a number of valuable suggestions and remarks and for \nhaving correted the mistakes in the first version of this work.\n\n\\bibliographystyle{nyjplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nLet $G\\subset \\mathrm{SU(2)}$ be a finite subgroup. According to McKay correspondence,\nsuch a subgroup gives rise to a graph $Q$ which turns out to be an affine\nDynkin diagram of ADE type. Let $\\g$ be the corresponding affine Lie algebra. \n\n\n\nThere are several approaches allowing one to construct $\\g$ from $G$: \n\\begin{enumerate}\n\\item There is a geometric construction, due to Nakajima and Lusztig, of $\\g$\n and its representations in terms of the quiver varieties associated to graph\n $Q$. These varieties are closely related to the moduli spaces of instantons\n on the resolution of singularities $\\widetilde{\\CC^2\/G}$. \n\n\\item There is an algebraic construction, due to Ringel \\cite{ringel2}, which\n allows one to get the universal enveloping algebra $U\\n_+$ of the positive\n part of $\\g$ as the Hall algebra of the category $\\Rep Q_h$, where $h$ is an\n orientation of $Q$ and $Q_h$ is the corresponding quiver. It was shown by \n Peng and Xiao \\cite{peng-xiao} that replacing $\\Rep Q_h$ by the\n quotient $R=D^b(\\Rep Q_h)\/T^2$ of the corresponding derived category, we\n can get a description of all of $\\g$. Different choices of orientation\n give rise to equivalent derived categories $R$, with equivalences given\n by Bernstein-Gelfand-Ponomarev reflection functors; in\n terms of $\\g$, these functors correspond to the action of the braid group\n of the corresponding Weyl group.\n \n \\end{enumerate}\nRecently, a third approach was suggested by Ocneanu \\cite{ocneanu}\n(unpublished), in a closely related setting of a subgroup in quantum\n$\\mathrm{SU(2)}$, for $q$ being a root of unity. His approach is based on\nstudying essential paths in ${\\widehat{Q}}=Q\\times \\Z\/h\\Z$, where $h$ is the\norder of the root of unity. This approach is purely combinatorial: all\nconstructions are done using this finite graph and vector spaces of\nessential paths between points in this graph, without involving any\ncategories at all. \n\nThis paper grew out of the author's attempt to understand Ocneanu's\nconstruction and in particular, find the appropriate\ncategorical interpretation of his combinatorial constructions;\nhowever, for simplicity we do it for subgroups in classical $\\mathrm{SU(2)}$,\nleaving the analysis of the subgroups in quantum $\\mathrm{SU(2)}$ for future\npapers. We show that Ocneanu's essential paths in ${\\widehat{Q}}$ have a\nnatural interpretation in terms of the category ${Coh_G(\\PP)}$ of\n$G$--equivariant $\\OO$-modules on $\\PP$ (or, rather,\nits ``even part'' $\\C={Coh_G(\\PP)}_0$, for certain natural $\\Z_2$ grading on\n${Coh_G(\\PP)}$). This also provides a relation with Ringel--Lusztig\nconstruction: for (almost) any choice of\norientation $h$ of $Q$, we construct an equivalence of triangulated\ncategories \n$$\nR\\Ph_h\\colon D^b( {Coh_G(\\PP)}_0)\\simeq D^b(\\Rep Q_h)\n$$\nThese equivalences agree with the equivalences of $D^b(\\Rep Q_h)$ for\ndifferent choices of $h$ given by reflection functors. As a corollary, we\nsee that the Grothendieck group $L=K(\\C)$ is an affine root lattice, and\nthe set $\\Delta$ of classes of indecomposable modules is an affine root\nsystem. \n\nThis construction of the affine root system via equivariant\nsheaves has a number of remarkable properties, namely:\n\\begin{enumerate}\n \\item This does not require a choice of orientation of\n $Q$ (unlike the category $D^b(\\Rep Q_h)$, where we first choose\n an orientation and then prove that the resulting\n derived category is independent of orientation).\n \\item The indecomposable objects in the category ${Coh_G(\\PP)}$ can be\n explicitly described. Namely, they are the\n sheaves $\\OO(n)\\ttt X_i$, where $X_i$ are irreducible\n representations of $G$ (these sheaves and their translations\n correspond to real roots of $\\g$) and\n torsion sheaves, whose support is a $G$--orbit in $\\PP$ (in\n particular, torsion sheaves whose support is an orbit of a\n generic point correspond to imaginary roots of $\\g$).\n \\item This construction of the affine root system does not give \n a natural polarization into negative and positive roots.\n Instead, it gives a canonical Coxeter element in the corresponding\n affine Weyl group, which is given by $C[\\F]=[\\F\\ttt \\OO(-2)]$\n (this also corresponds to the Auslander--Reiten functor\n $\\tau$).\n \n \\item This construction gives a bijection of the vertices of the\n affine Dynkin diagram $Q$ and (some of) the $C$-orbits in\n $\\Delta$ (as opposed to the quiver construction, where vertices\n of $Q$ are in bijection with the simple positive roots). \n \n\\end{enumerate}\n\n\n\nFor $G=\\{1\\}$ (which is essentially equivalent to $G=\\{\\pm 1\\}$),\nthese results were first obtained in the paper \\cite{baumann-kassel},\nwhere it was shown that the corresponding Hall algebra contains the\nsubalgebra isomorphic to $U\\widehat{\\slt}$. This paper in turn was\ninspired by an earlier paper of Kapranov \\cite{kapranov}. \n\n\nIt should be noted that many of the results obtained here have already\nbeen proved in other ways. Most importantly, it had been proved by\nLenzing \\cite{geigle-lenzing,lenzing} that the derived category of\nequivariant sheaves on $\\PP$ is equivalent to the derived category of\nrepresentation of the corresponding quiver; this result has been used by\nSchiffmann \\cite{schiffmann,schiffmann2} for construction of a subalgebra\nin the corresponding affine Lie algebra via Hall algebra of the category of\nequivariant sheaves. However, the construction of\nequivalence in \\cite{lenzing} is different than the one suggested here. The\nprimary difference is that in Lenzing's construction, the Dynkin diagram\n$Q$ is constructed as the star diagram, with lengths of branches determined\nby the branching points of the cover $\\PP\\to X=\\PP\/G$, and he uses\na standard orientation of this diagram. In the construction presented\nhere, the diagram is $Q$ is defined in a more standard way, using the set\n$I$ of irreducible representations of $G$; more importantly, we construct\nnot a single equivalence but a collection of equivalences, one for each\nadmissible orientation. In Lenzing's construction, torsion sheaves\nnaturally play a major role; in our construction, we concentrate on\nthe study of locally free sheaves. \n\nLenzing's results apply not only to $\\PP\/G$ but to a much larger\nclass of ``non-commutative projective curves''. However, the downside of\nthis is that the language he uses is rather technical, making his papers\nsomewhat hard for non-experts. For this reason, we had chosen to give\nindependent proofs of some of the results, thus saving the reader the\nnecessity of learning the language of non-commutative curves. Of course,\nwe tried to clearly mark the results which had already been known.\n\n\n{\\bf Acknowledgments.} The author would like to thank P.~Etingof,\nV.~Ostrik, and O.~Schiffmann for fruitful discussions. Special thanks to\nA.~Ocneanu, whose talk inspired this paper. \n\n\\section{Basic setup}\\label{s:basic}\nThroughout the paper, we work over the base field $\\CC$ of complex\nnumbers. $G$ is a finite subgroup in $\\mathrm{SU(2)}$; for simplicity, we assume\nthat $\\pm I\\in G$ and denote $\\Gbar=G\/\\{\\pm I\\}$ (this excludes\n$G=\\Z_n$, $n$ odd; this case can also be included, but some of the\nconstructions of this paper will require minor changes). We denote by\n$V$ the standard 2-dimensional space, considered as a tautological\nrepresentation of $\\mathrm{SU(2)}$ (and thus of $G$) and \n$$\nV_k=S^k V\n$$\nis the $k+1$--dimensional irreducible representation of $\\mathrm{SU(2)}$ (and\nthus a representation, not necessarily irreducible, of $G$). \n\nWe denote by $\\Rep G$ the category of finite-dimensional\nrepresentations of $G$ and by $I=\\Irr G$ the set of isomorphism\nclasses of simple representations; for $i\\in I$, we denote by $X_i$\nthe corresponding representation of $G$. Since $G\\supset \\{\\pm I\\}$,\nthe category $\\Rep G$ is naturally $\\Z_2$--graded:\n\\begin{equation}\\label{e:even-odd}\n\\Rep G=\\Rep_0 G\\oplus \\Rep_1 G,\\quad \n\\Rep_p G=\\{V\\in \\Rep G\\st (-I)|_V=(-1)^p\\id \\}\n\\end{equation}\nFor homogeneous object $X$, we define its ``parity'' $p(X)\\in \\Z_2$\nby \n\\begin{equation}\\label{e:parity}\np(X)=p\\text{ if } X\\in \\Rep_p G\n\\end{equation}\nin particular, $p(V)=1$. We will also define, for $i\\in I$, its\nparity $p(i)=p(X_i)$. This gives a decomposition \n\\begin{equation}\\label{e:decomposition_I}\nI=I_0 \\sqcup I_1.\n\\end{equation}\n\n\nWe define the graph $Q$ with the set of vertices $V(Q)=I$ and\nfor every two vertices $i,j$, the number of edges connecting them is\ndefined by \n$$\nn(i,j)=\\dim \\Hom_G(X_i,X_j\\ttt V)\n$$\n\nNote that decomposition \\eqref{e:decomposition_I} shows that this\ngraph is bipartite: $V(Q)=V_0(Q)\\sqcup V_1(Q)$, and all edges connect\nvertices of different parities. \n\n\nIt is well-known that one can construct an isomorphism\n$$\n\\{\\text{Paths of length $l$ in $Q$ connecting $i,j$}\\}\n=\\Hom_G(X_i, V^{\\ttt l}\\ttt X_j)\n$$\nand that $\\Hom_G(X_i, V_l \\ttt X_j)$ can be described as the space of\n``essential paths'' in $Q$, which is naturally a direct summand in\nthe space of all paths (see \\cite{coq-garcia}). The algebra of essential\npaths is also known as the preprojective algebra of $Q$. \n\n\nBy McKay correspondence, $Q$ must be an affine Dynkin diagram. \nWe denote by $\\Delta(Q)$ the corresponding affine root system; it has a\nbasis of simple roots $\\alpha_i, i\\in I$. We denote \n$$\nL(Q)=\\bigoplus_{i\\in I} \\Z\\alpha_i =\\Z^I\n$$\nthe corresponding root lattice. It has a natural bilinear\nform given by $(\\alpha_i,\\alpha_i)=2$ and $(\\alpha_i,\\alpha_j)=-n(i,j)$; as is\nwell-known, this form is positive semidefinite. The kernel of this\nform is $\\Z\\delta$, where $\\delta$ is the imaginary root of $\\Delta(Q)$. \n\nWe denote by $s_i\\colon L(Q)\\to L(Q)$ the reflection around root\n$\\alpha_i$. \n\nFinally, let $K(G)$ be the Grothendieck group of the category $\\Rep G$.\nIt is freely generated by classes $[X_i], i\\in I$; thus, we have a\nnatural isomorphism \n\\begin{equation}\\label{e:K(G)}\n\\begin{aligned}\nK(G)&\\isoto L(Q)\\\\\n[X_i]&\\mapsto \\alpha_i\n\\end{aligned}\n\\end{equation}\n\n\n\n\\section{Quiver $Q_h$}\n\nWe will consider a special class of orientations of $Q$. \n\n\\begin{definition}\\label{d:height_function}\nA {\\em height function} $h$ is a map $I\\to \\Z$ such that \n$h(i)-h(j)=\\pm 1$ if $i,j$ are connected by an edge in $Q$, and\n$h(i)\\equiv p(i) \\pmod 2$. \n\\end{definition}\n\nEvery height function gives rise to orientation of edges of $Q$: if\n$h(j)=h(i)-1$ then all edges connecting $i$ and $j$ are directed\ntowards $j$:\n$$\ni\\longrightarrow j \\quad\\text{ if } h(j)=h(i)-1\n$$\n\n\nWe will denote by $Q_h$ the quiver given by this orientation. We will\nwrite $i\\to j$ if there exists an edge whose tail is $i$ and head is\n$j$. The notation $\\sum_{j\\colon j\\to i}$ will mean the sum over all\nvertices $j$ connected with $i$ by an edge $j\\to i$; if there are\nmultiple edges, the corresponding vertex $j$ will be taken more than\nonce. \n\nOrientations obtained in this way will be called {\\em\nadmissible} (note: our use of this word is slightly different from the\nuse in other sources). It is easy to check that if $Q$ has no loops,\nthen any orientation of $Q$ is admissible. For type $A$, an\norientation is admissible if the total number of clockwise arrows is\nequal to the number of counterclockwise ones (which again rules out\ntype $\\widehat{A}_n$, $n$ even, corresponding to $G=\\Z_{n+1}$). It is\nalso obvious that $Q_h$ has no oriented loops, and that\nadding a constant to $h$ gives the same orientation. \n\nGiven a height function $h$, we will draw $Q$ in the plane so that\n$h$ is the $y$--coordinate; then all edges of $Q_h$ are directed\ndown. \\firef{f:Qh} shows an example of a height function for\nquiver of type $D$.\n\\begin{figure}[ht]\n\\fig{Qh.eps}\n\\caption{Example of a height function for Dynkin diagram\n$\\widehat{D}_7$}\\label{f:Qh}\n\\end{figure}\n\n\\begin{definition}\\label{d:reflection_functors1}\nLet $h$ be a height function on $Q$, and $i\\in I$ be a sink in $Q_h$:\nthere are no edges of the form $i\\to j$ (in terms of $h$, it is\nequivalent to saying that $h$ has a local minimum at $i$). We define\nnew height function \n$$\ns_i^+h (j)=\\begin{cases}\n h(j)+2, & j=i\\\\\n h(j), & j\\ne\\i\n \\end{cases}\n$$\nSimilarly, if $i$ is a source, i.e., there are no vertices of the\nform $j\\to i$ (equivalently, $h$ has a local maximum at $i$), then \nwe define \n$$\ns_i^-h (j)=\\begin{cases}\n h(j)-2, & j=i\\\\\n h(j), & j\\ne\\i\n \\end{cases}\n$$\n\\end{definition}\nOne easily sees that $s_i^+h$, $s_i^-h$ are again height functions;\nthe quiver $Q_{s_i^\\pm h}$ is obtained from $Q_h$ by reversing\norientation of all edges adjacent to $i$. We will refer to $s_i^\\pm$\nas (elementary) orientation reversal operations. \n\nThe following lemma is known; however, for the benefit of the reader\nwe included the proof. \n\n\\begin{lemma}\\label{l:reflection_functors1}\nAny two height functions $h, h'$ can be obtained one from another by\na sequence of orientation reversal operations $s_i^\\pm$. \n\\end{lemma}\n\\begin{proof}\nDefine the ``distance'' between two height functions by \\\\\n$d(h,h')=\\sum_I |h(i)-h'(i)|\\in \\Z_+$. We will show that if $d(h,h')>0$, then\none can\nfind $s_i^\\pm$ such that $s_i^\\pm$ can be applied to $h$, and\n$d(s_i^\\pm h, h')h'(i)\\}$. One easily sees that if $i\\in I_+$, and\n$j\\to i$ in $Q_h$, then $j\\in I_+$. Thus, if $I_+$ is non-empty, it \nmust contain at least one source $i$ for $Q_h$. But then $d(s_i^- h,\nh')=d(h, h')-2$. \n\nSimilarly, let $I_-=\\{i\\st h(i)1. \n$$\n\n For a representation $M=(M_i)_{i\\in I}$ of $Q_h$, we define its\ndimension $\\dim X\\in L(Q)$ by $\\dim M=\\sum (\\dim M_i)\\alpha_i$ (recall that\n$L(Q)=\\Z[I]$ is the root lattice of the root system $\\Delta(Q)$, see\n\\seref{s:basic}). \n \nThe following theorem summarizes some of the known results about\nrepresentations of quivers and root system $\\Delta(Q)$. \n\n\n\\begin{theorem}\\label{t:reps_of_quivers}\n\\par\\indent\n\\begin{enumerate}\n\\item The map $[X]\\mapsto \\dim X$ gives an isomorphism $K(Q_h)\\isoto\nL(Q)$. Under this isomorphism, the bilinear form on $L(Q)$ is identified\nwith the following bilinear form on $K(Q_h)$: \n $$\n (x,y)=\\+\\\n $$\n where by definition \n $$\n \\<[X],[Y]\\>= \\dim \\RHom (X,Y)=\\dim \\Hom (X,Y) -\\dim \\Ext^1(X,Y).\n $$\n \\item The set of dimensions of indecomposable modules is exactly the\n set $\\Delta_+(Q)$ of positive roots in $\\Delta(Q)$. For real roots\n $\\alpha$, there is exactly one up to isomorphism indecomposable\n module $M_\\alpha$ of dimension $\\alpha$; for\n imaginary root $\\alpha$, there are infinitely many pairwise\n non-isomorphic modules of dimension $\\alpha$. \n \n\\end{enumerate}\n\\end{theorem}\n\nThere is also an explicit description of indecomposables in\n$\\D^b(Q_h)$ (see \\cite[Lemma I.5.2]{happel}). \n\\begin{theorem}\\label{t:indecompos_derived}\n Indecomposable objects in $\\D^b(Q_h)$ are of the form $M[n]$, where\n $M$ is an indecomposable object in $\\Rep Q_h$, $n\\in \\Z$. \n\\end{theorem}\n\n\nWe will also need reflection functors of Bernstein--Gelfand--Ponomarev.\nRecall that if $i$ is a sink in $Q_h$, then one has a natural functor \n\\begin{equation}\\label{e:reflection_2}\n S_i^+\\colon \\Rep(Q_h)\\to \\Rep(Q_{s_i^+h})\n\\end{equation}\nsimilarly, if $i$ is a source, one has a natural functor \n\\begin{equation}\\label{e:reflection_3}\n S_i^-\\colon \\Rep(Q_h)\\to \\Rep(Q_{s_i^-h})\n\\end{equation}\n(see definition in \\cite{drab-ringel},\n\\cite{kraft-riedtmann}). \n\nThe following result is known and easy to prove, so we skip the\nproofs. \n\\begin{theorem}\\label{t:coxeter2}\n\\par\\indent \n \\begin{enumerate}\n \\item Functor $S_i+$ is left exact and $S_i^-$ is right exact. \n \n \n We will denote by $RS_i^+, LS_i^-\\colon \\D^b(Q_h)\\to\n \\D^b(Q_{s_i^\\pm h})$ the corresponding derived functors. \n \\item\n The functors $RS_i^+$, $LS_i^-$ are equivalences of categories\n $\\D^b(Q_h)\\to\\D^b(Q_{s_i^\\pm h})$, which induce the usual\n reflections $s_i$ on the Grothendieck group: \n $$\n \\dim RS_i^+ (X)=s_i(\\dim X), \\quad \\dim LS_i^- (X)=s_i(\\dim X)\n $$\n \\item If $i,j$ are not neighbors in $Q$, then $RS_i^+,\n RS_j^+$ commute (i.e., compositions in different orders are\n isomorphic) and similarly for $LS_i^-$. \n \\end{enumerate}\n\\end{theorem}\n\n\n\nIn particular, for a given height function $h$ let $s^+_{i_1}\\dots\ns^+_{i_r}$ be a sequence of elementary orientation\nreversal operations such that \n$$\ns_{i_1}^+\\dots s_{i_r}^+ (h) =h+2\n$$\nOne easily sees that this condition is equivalent to requiring that every\nindex $i\\in I$ appears in the sequence $\\{i_1,\\dots, i_r\\}$ exactly once;\nit follows from \\leref{l:reflection_functors1} that for every height\nfunction $h$, such sequences of orientation reversal operations exist. \nFor such a sequence, the corresponding element of the Weyl group\n\\begin{equation}\\label{e:coxeter}\nc^+_h=s_{i_1}\\dots s_{i_r}\n\\end{equation}\nis called the {\\em Coxeter element}, and\nthe corresponding composition of reflection functors \n\\begin{equation}\\label{e:C+}\nRC^+_h=RS_{i_1}^+\\dots RS_{i_r}^+\\colon \\D^b(Q_h)\\to\\D^b(Q_{h+2})\n\\end{equation}\nwill be called the {\\em Coxeter functor}. Note that since $Q_h\\simeq\nQ_{h+2}$ as a quiver, we can consider$RC_h^+$ as an autoequivalence of\n$\\D^b(Q_h)$. \n\nIt is easy to show (see \\cite{drab-ringel}, \\cite{shi}) that the\nCoxeter element $c^+_h$ only depends on $h$ and not on the choice of\nthe sequence $i_1,\\dots,i_r$; moreover, the proof of this only uses\nthe fact that $s_i, s_j$ commute if $i,j$ are not connected in $Q$\nand does not use the braid relations. Thus, by \\thref{t:coxeter2},\nthis implies that up to an isomorphism, $RC^+_h$ also depends only on\n$h$; this justifies the notation $RC_h^+$.\n\nSimilarly, we can define functors \n\\begin{equation}\\label{e:coxeter2}\nLC^-_h\\colon \\D^b(Q_h)\\to\\D^b(Q_{h-2})\\simeq\\D^b(Q_h)\n\\end{equation}\nusing sequences of orientation reversals $s_{i_1}^-\\dots s_{i_r}^- h\n=h-2$; the corresponding element of the Weyl group will be denoted by\n$c_h^-$. As before, it can be shown that $LC_h^-$, $c_h^-$ only depend\non $h$\n\n\nFor readers familiar with the theory of Auslander--Reiten sequences\n(see \\cite{auslander-reiten}, \\cite{happel}), we add\nthat the category $\\Rep Q_h$ has Auslander--Reiten sequences, and the\nAuslander--Reiten functor $\\tau$ is given by $\\tau=C_h^-$.\n\n\n\\section{Equivariant sheaves}\nIn this section, we introduce the main object of this paper, the\ncategory of equivariant sheaves on $\\PP$. Most of the results of\nthis section are well-known and given here only for the convenience\nof references. \\leref{l:frobenius} does not seem to be easily\navailable in the literature, but is very easy to prove. \n\nLet $V^*$ be the dual of the tautological representation $V$ of\n$\\mathrm{SU(2)}$. Since $G$ is a finite subgroup in $\\mathrm{SU(2)}$, it naturally acts on\n$\\PP=\\mathbb{P}(V^*)$, and the structure sheaf $\\OO$ has a standard\n$\\mathrm{SU(2)}$- (and thus $G$-) equivariant structure. Moreover, all twisted\nsheaves $\\OO(n)$ also have a standard equivariant structure, so that\nthe space of global sections $\\Gamma(\\OO(n))$ is a representation of \n$G$: \n\\begin{equation}\\label{e:H0}\n\\Gamma(\\OO(n))=\\begin{cases}\n S^nV=V_n, &n\\ge 0\\\\\n 0, &n<0\n \\end{cases}\n\\end{equation}\nSimilarly, the higher homology spaces are naturally representations of $G$: it\nis well-known that $H^i(\\PP, \\OO(n))=0$ for $i>1$, and \n\\begin{equation}\\label{e:H1}\nH^1(\\PP,\\OO(n))=\\begin{cases}\n S^{-n-2}V^*=V^*_{-n-2}, &n\\le -2\\\\\n 0, &n\\ge -1\n \\end{cases}\n\\end{equation}\n\nLet ${Qcoh_G(\\PP)}$, ${Coh_G(\\PP)}$ be the categories of $G$--equivariant\nquasi-coherent (respectively, coherent) $\\OO$--modules on $\\PP$\n(see, e.g., \\cite[Section 4]{bkr} for definitions). Note that we are\nconsidering isomorphisms $\\lambda_g\\colon \\F\\to g^*\\F$ as part of the\nstructure of the $G$--equivariant sheaf. For brevity, we will denote \nmorphisms and $\\Ext$ groups in ${Qcoh_G(\\PP)}$ by $\\Hom_G(X,Y)$, $\\Ext_G(X,Y)$.\nFor an equivariant sheaf $\\F$ we will denote \n$$\n\\F(n)=\\OO(n)\\ttt_\\OO \\F\n$$\nwith the obvious equivariant structure. Similarly, for a\nfinite-dimensional representation $X$ of $G$, we denote \n$$\nX(n)=\\OO(n)\\ttt_\\CC X.\n$$\n\n\nWe list here some of the basic properties of equivariant\nsheaves; proofs can be found in \\cite[Section 4]{bkr}.\n\\begin{theorem}\\label{t:Gcoh}\n\\par\\noindent\n \\begin{enumerate}\n \\item ${Qcoh_G(\\PP)}$ is an abelian category, and a sequence $0\\to\n \\F_1\\to \\F_2\\to \\F_3\\to 0$ is exact in ${Qcoh_G(\\PP)}$ iff it is exact in\n $Qcoh(\\PP)$. \n\n \\item For any $\\F,\\G\\in {Qcoh_G(\\PP)}$, the space $\\Hom_\\OO(\\F,\\G)$ is a\n representation of $G$, and $\\Hom_G(\\F,\\G)=(\\Hom_\\OO(\\F,\\G))^G$.\n Similarly, \\\\\n $\\Ext_G^i(\\F,\\G)=(\\Ext^i_\\OO(\\F,\\G))^G$; in\n particular, $\\Ext_G^i(\\F, \\G)=0$ for $i>1$.\n \n \\item For any $\\F,\\G\\in {Coh_G(\\PP)}$, the spaces $\\Hom_G(\\F,\\G)$,\n $\\Ext_G^1(\\F,\\G)$ are finite-dimensional. \n\n \\item For any $\\F,\\G\\in {Coh_G(\\PP)}$, one has \n \\begin{align*}\n \\Hom_G(\\F, \\G(n))=\\Hom_G(\\F(-n),\\G)=0\\quad \\text{for }n\\ll 0,\\\\\n \\Ext_G^1(\\F, \\G(n))=\\Hom_G(\\F(-n),\\G)=0\\quad \\text{for }n\\gg 0,\n \\end{align*}\n \n\\end{enumerate}\n\\end{theorem}\n\n\nAs an immediate corollary of this, we see that the category ${Coh_G(\\PP)}$ has the \nKrull--Schmidt property: every object $\\F\\in {Coh_G(\\PP)}$ can be written as a direct\nsum of indecomposable modules, and the multiplicities do not depend on the\nchoice of such decomposition. It is also a hereditary category (recall that a\ncategory is called hereditary if $\\Ext^2(A,B)=0$ for any objects $A,B$; in\nparticular, this implies that a quotient of an injective object is injective,\nas can be easily seen from the long exact sequence of $\\Ext$ groups). \n\nWe say that $\\F\\in{Coh_G(\\PP)}$ is locally free if it is locally free as an\n$\\OO$-module. \n\\begin{theorem}\\label{t:locally_free}\n\\par\\noindent\n\\begin{enumerate}\n \\item Every locally free $\\F\\in{Coh_G(\\PP)}$ is isomorphic to direct sum of\n sheaves of the form \n \\begin{equation}\n X_i(n)=\\OO(n)\\ttt X_i, \\quad X_i \\text{ -- irreducible representation of\n$G$} \\end{equation}\n \\item If $\\F\\in{Coh_G(\\PP)}$ is locally free, then the functor $\\ttt \\F\\colon\n {Coh_G(\\PP)}\\to {Coh_G(\\PP)}$ is exact. \n \n \\item Every coherent $G$-equivariant sheaf has a resolution consisting of\n locally free sheaves. \n \\item \\textup{(}Serre Duality\\textup{)} \n For any two locally free sheaves $\\F, \\G$ we have isomorphisms \n $$\n \\Ext_G^1(\\F, \\G(-2))=\\Ext^1(\\F(2),\\G)=\\Hom_G(\\G, \\F)^* \n $$\n\\end{enumerate}\n\\end{theorem}\n\nIt immediately follows from computation of homology of $\\OO(n)$ as coherent\nsheaves that \n\\begin{equation}\\label{e:hom1}\n \\Hom_G ( X_i(n), X_j(k))\n =\\begin{cases}\n \\Hom_G( X_i, V_{k-n}\\ttt X_j), &k\\ge n\\\\\n 0, & kh_i \\iff (q\\notin Q_h, q\\succ q'$ for some $q'\\in Q_h)$. In\n this case, we will say that $q$ is above $Q_h$ and write\n $q\\succ Q_h$.\n \\item $n1$, and \n$$\n(R^1\\Ph_h(\\F))(i)=\\Ext_\\C^1(X_i(h_i),\\F)\n$$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFollows from left exactness of functor $\\Hom_\\C(X_i(h_i), -)$ and definitions. \n\\end{proof}\n\n\n\\begin{example}\\label{x:standard}\nLet $h$ be the ``standard'' height function: $h(i)=0$ for $i\\in I_0$,\n$h(i)=1$ for $i\\in I_1$. Then the map $\\Ph_h$ can also be described\nas follows. Consider the functor $\\Rep Q_h\\to \\Rep G$ defined by\n$\\{V_i\\}\\mapsto \\bigoplus V_i\\ttt X_i$. Then the composition\n$$\n\\C\\xxto{\\Ph_h} \\Rep Q_h\\to \\Rep G\n$$\nis given by \n$$\n\\F\\mapsto \\Gamma(\\F)\\oplus \\Gamma(\\F(-1)).\n$$\n\nIndeed, for each representation $M$ of $G$ we have \n$$\nM\\simeq\\bigoplus \\Hom_G(X_i,M)\\ttt X_i\n$$\nApplying it to $\\Gamma(\\F)=\\Hom_\\OO(\\OO,\\F)$ we get \n$$\n\\Gamma(\\F)=\\bigoplus_{i\\in I} \\Hom_\\OO(X_i\\ttt \\OO,\\F)^G\\ttt X_i\n=\\bigoplus_{i\\in I} \\Hom_G (X_i\\ttt \\OO,\\F)\\ttt X_i\n$$\nSince $\\F\\in \\C={Coh_G(\\PP)}_0$, we have $\\Hom_G (X_i\\ttt \\OO,\\F)=0$ for\n$i\\in I_1$; thus, \n$$\n\\Gamma(\\F)=\\bigoplus_{i\\in I_0} \\Hom_G (X_i\\ttt \\OO,\\F)\\ttt X_i\n$$\nSimilar argument shows that \n$$\n\\Gamma(\\F(-1))=\\bigoplus_{i\\in I_1} \\Hom_G (X_i\\ttt \\OO(1),\\F)\\ttt X_i\n$$\nThus, \n$$\n\\Gamma(\\F)\\oplus \\Gamma(\\F(-1))=\n\\left(\\bigoplus_{i\\in I_0} \\Hom_\\C (X_i,\\F)\\ttt X_i\\right)\n\\oplus\n\\left(\\bigoplus_{i\\in I_1} \\Hom_\\C (X_i(1),\\F)\\ttt X_i\\right)\n$$\nas desired. \n\\end{example}\n\nSince $\\Ph_h$ is left exact, we can define the associated derived functor\n$R\\Ph_h\\colon \\D^b(\\C)\\to \\D^b(\\Rep Q_h)$. \nThe following two theorems are the main results of this paper. \n\n\\begin{theorem}\\label{t:main2}\nFor any height function $h$, the functor $R\\Ph_h\\colon \\D^b(\\C)\\to\n\\D^b(Q_h)$ defined by \\eqref{e:Ph} is an equivalence of triangulated\ncategories.\n\\end{theorem}\n\\begin{proof}\nIt follows from \\leref{l:vanishing_of_homs} that the object \n$$\nT=\\bigoplus_{q\\in Q_h}X_q\n$$\nsatisfies $\\Ext^1(T,T)=0$. By \\leref{l:generators}, direct summands of\n $T$ generate $\\D^b(\\C)$ as a triangulated category. Therefore, $T$ is\nthe tilting object in $\\D^b(\\C)$ as defined in \\cite{geigle-lenzing}.\nNow the statement of the theorem follows from the general result of\n\\cite{geigle-lenzing}.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{t:main1}\nLet $i$ be a sink for $h$, and $S_i^+$ the reflection functor defined by \n\\eqref{e:reflection_2}. Then the following diagram is commutative:\n$$\n\\xymatrixrowsep{4pt}\n\\xymatrixcolsep{40pt}\n\\xymatrix{\n &\\D^b(Q_{s_i^+h})\\\\\n\\D^b(\\C)\\ar[ur]^{R\\Ph_{s_i^+h}}\\ar[dr]^{R\\Ph_{h}}&\\\\\n &\\D^b(Q_{h})\\ar[uu]_{RS_i^+}\n} \n$$\nand similarly for $S_i^-$. \n\\end{theorem}\n\\begin{proof}\n\n\n\nLet us first prove that if $q\\succ Q_h$, then \n$$\nR\\Ph_{s_i^+h}(X_q)=RS_i^+\\circ R\\Ph_h (X_q).\n$$ \nIndeed, in this case it follows\nfrom \\leref{l:Ph}, \\leref{l:vanishing_of_homs} that $R^i\\Ph_h(X_q)=0$\nfor $i>0$,\nso $R\\Ph_h(X_q)=\\Ph_h(X_q)$; similarly, \n$R\\Ph_{s_i^+h}(X_q)=\\Ph_{s_i^+h}(X_q)$.\n\nLet $i$ be a sink for $Q_h$. Then we have the Auslander--Reiten exact\nsequence\n$$\n0\\to X_i(h_i)\\to \\bigoplus_{j} X_j(h_j)\\to X_i(h_i+2)\\to 0\n$$\nApplying to this sequence functor $\\Hom_\\C(-, X_q)$ and using\n\\leref{l:vanishing_of_homs} which gives vanishing of\n$\\Ext^1$, we get a short exact\nsequence \n$$\n0\\to \\Hom_\\C(X_i(h_i+2), X_q)\\to \\bigoplus_{j} \\Hom_\\C(X_j(h_j), X_q)\\to\n\\Hom_\\C(X_i(h_i), X_q)\\to 0\n$$\nComparing this with definition of $S_i^+$, we see that \n$\\Ph_{s_i^+h}(X_q)=S_i^+(\\Ph_h(X_q))$. \n\nThus, we have shown that for $q\\succ Q_h$, we\nhave $R\\Ph_{s_i^+h}(X_q)=RS_i^+\\circ R\\Ph_h (X_q)$. \n\n\nTo complete the proof, we now use the following easy result.\n\n\\begin{lemma}\\label{l:technical4}\n Let $\\D$, $\\D'$ be triangulated categories, $\\Ph_1,\\Ph_2\\colon\n \\D\\to \\D'$ triangle functors, and $\\alpha\\colon\n \\Ph_1\\to\\Ph_2$ a morphism of functors. Assume that there is a\n collection of objects $X_q\\in \\D$ which generate $\\D$ as a\n triangulated category in the sense of \\leref{l:generators} and such\n that for each $X_q$, \n $$\n \\alpha\\colon \\Ph_1(X_q)\\to \\Ph_2(X_q)\n $$\n is an isomorphism. Then $\\alpha$ is an isomorphism of functors. \n\\end{lemma} \n\nApplying this lemma to $\\D=\\D^b(\\C)$, $\\D'=\\D^b(Q_{s_i^+h})$,\n$\\Ph_1=R\\Ph_{s_i^+ h}$, $\\Ph_2=RC_i^+\\circ R\\Ph_h$ and the \ncollection of objects $X_q, q\\succ Q_h$ (which\ngenerate $\\D^b(\\C)$ by \\leref{l:generators}), we get the statement of\nthe theorem. \n\\end{proof}\n\n\n\\begin{corollary}\\label{c:coxeter}\nWe have the following commutative diagram\n$$\n\\xymatrixrowsep{20pt}\n\\xymatrixcolsep{30pt}\n\\xymatrix{\n \\D^b(\\C)\\ar[r]^-{R\\Ph_h} \n &\\D^b(Q_h)\\simeq \\D^b(Q_{h+2})\\\\\n \\D^b(\\C)\\ar[r]^{R\\Ph_h}\\ar[u]_{\\ttt \\OO(-2)} \n &\\D^b(Q_h)\\ar[u]_{RC_h^+}\n}\n$$\nwhere $RC_h^+$ is the Coxeter element \\eqref{e:C+}, and similarly for\n$LC_h^-$. \n\\end{corollary}\n\n\n\\begin{remark}\nIt should be noted that while $R\\Ph_h$ is an equivalence of derived\ncategories, it is definitely not true that $\\Ph_h$ is an equivalence\nof abelian categories. For example, there are no injective or\nprojective objects in $\\C$ while there are enough injectives and\nprojectives in $\\Rep Q_h$. Similarly, the set of simple modules is\nvery much different in these two categories. However, the sets of\nindecomposable modules are effectively the same. \n\\end{remark}\n\n\n\\section{The root system and the Grothendieck group}\nIn this section, we list some important corollaries of the equivalence of\ncategories constructed in the previous section. Again, many of these\nresults can also be obtained form the equivalence of categories\nconstructed by Lenzing; however, we choose to provide an independent\nexposition. \n\n\nThoroughout this section, we let $L=K(\\C)$ be the Grothendieck group of\ncategory $\\C$. We define the inner product on $L$ by \n\\begin{equation}\\label{e:euler_form}\n (x,y)=\\+\\\n\\end{equation}\n where by definition \n$$\n \\<[X],[Y]\\>=\\dim \\RHom (X,Y)=\\dim \\Hom (X,Y) -\\dim \\Ext^1(X,Y)\n$$\n(compare with \\thref{t:reps_of_quivers}). We define $\\Delta\\subset L$ by \n\\begin{equation}\n\\Delta=\\{[X], X \\text{--- a non-zero indecomposable object in\n}\\D^b(\\C)\\}\n\\end{equation}\n\nFinally, we define the map $C\\colon \\Delta\\to\\Delta$ by \n$$\nC([X])=[X(-2)]\n$$\n\n\n\n\\begin{theorem}\\label{t:main3}\n\\par\\indent\n\\begin{enumerate}\n \\item The set $\\Delta\\subset L$ is an affine root system, and $C$ is\n a Coxeter element. \n \\item Recall the lattice $L(Q)$ and root system $\\Delta(Q)$ from\n \\seref{s:basic}. Then for any height function $h$ on $Q$, the map\n \\begin{equation}\n \\begin{aligned}\n R\\Ph_h\\colon L&\\to L(Q)\\\\\n [\\F]&\\mapsto \\bigoplus \n \\biggl(\\dim \\RHom_\\C(X_i(h_i), \\F)\\biggr) \\alpha_i\n \\end{aligned}\n \\end{equation}\n is an isomorphism of abelian groups which identifies\n $\\Delta\\subset L$ with $\\Delta(Q)\\subset L(Q)$ and $C$ with the\n Coxeter element $c_h^+$ defined by \\eqref{e:coxeter}. \n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n The first part follows from the second one; the\n second part follows from the fact that $R\\Ph_h$ is an equivalence of\n categories (\\thref{t:main2} and \\coref{c:coxeter}). \n\\end{proof}\n\n\n\n\\begin{corollary}\\label{c:generators_of_K(C)}\n \\par\\noindent\n \\begin{enumerate}\n \\item \n For any height function $h$, the classes $[X_q]$, $q\\in Q_h$, are free\n generators of the Grothendieck group $K(\\C)$. \n \\item \n $K(\\C)$ is generated by the classes $[X_i(n)], n\\equiv p(i)\\mod 2$,\n and AR relations \\eqref{e:aus-reiten3} is the full list of all\n relations among classes $[X_i(n)]$.\n \\end{enumerate}\n\\end{corollary}\n\\begin{proof}\nBy \\leref{l:generators}, classes $[X_q]$, $q\\in Q_h$, generate $K(\\C)$. On\nthe other hand, by \\thref{t:main3}, $K(\\C)$ is isomorphic to $\\Z^I$ and\nthus has rank $|I|$, so these generators must be linearly independent. \n\nTo prove the second part, denote temporarily $L'=Span([X_i(n)])\/J$, where\n$J$ is the subgroup generated by AR relations \\eqref{e:aus-reiten3}.\nSince $[X_i(n)]$ generate $K(\\C)$ and AR relations hold in $K(\\C)$, we\nsee that $K(\\C)$ is a quotient of $L'$. \n\nNow choose some height function $h$. It follows from \n\\leref{l:reflection_functors1} that $[X_q]$, $q\\in Q_h$, generate $L'$, so\nit has rank at most $|I|$. On the other hand, $K(\\C)$ has rank $|I|$.\nThus, $L'$ has rank $|I|$ and $L'=K(\\C)$. \n\\end{proof}\n\n\n\\begin{example}\\label{x:standard2}\n Let $h$ be the ``standard'' height function as defined in\n\\exref{x:standard}. Then the map\n$R\\Ph_h\\colon K(\\C)\\to L(Q)\\simeq K(G)$\n is given by \n \\begin{align*}\n &[X_i]\\mapsto \\alpha_i, \\quad i \\in I_0\\\\\n &[X_i(1)]\\mapsto \\alpha_i+\\sum_{j-i}\\alpha_j, \\quad i \\in I_1\n \\end{align*}\n and the corresponding Coxeter element is $C=(\\prod_{i\\in\n I_1}s_i)(\\prod_{i\\in I_0}s_i)$. \n \n This also implies that for this $h$, $[X_i(-1)]\\mapsto -\\alpha_i$, $i\\in\n I_1$; in particular, we see that classes\n $\\alpha_i, i\\in I_0$, and $-\\alpha_i, i\\in I_1$, form a set of representatives\n of $C$-orbits on on ${\\widehat{Q}}$. An analogous statement for finite root\n system has been proved in \\cite{kostant2}. \n\\end{example} \n\n\n\nFor completeness, we also describe here the indecomposable sheaves\ncorresponding to imaginary roots of $\\Delta$.\n\n\n\\begin{theorem}\\label{t:imaginary_root}\nLet $x\\in \\PP$ be a generic point: $\\Stab_\\Gbar x=\\{1\\}$, and let \n$$\n\\delta=[\\OO_{[Gx]}]\\in \\Delta\n$$\n\\textup{(}see \\eqref{e:torsion_sheaf}\\textup{)}. Then: \n\\begin{enumerate}\n\\item $\\delta$ does not depend on the choice of point $x$\n\\item $\\delta=\\delta_0-\\delta_1$, where \n\\begin{align*}\n\\delta_0&=\\sum_{i\\in I_0}d_i[X_i]=[R_0]\\\\\n\\delta_1&=\\sum_{i\\in I_1}d_i[X_i(-1)]=[R_1(-1)],\n\\end{align*}\n where $d_i=\\dim X_i$ and $R_0, R_1$ are even and odd parts of the\nregular representation defined by \\eqref{e:R_p}. \n\n\\item $C\\delta=\\delta$; $C\\delta_0=\\delta_0-2\\delta$; $C\\delta_1=\\delta_1-2\\delta$\n\\item For any $\\alpha\\in L$, $(\\delta,\\alpha)=0$\n\\item $\\delta$ is a generator of the set of imaginary roots of $\\Delta$:\n$$\n\\Delta^{im}=\\Z\\delta\n$$\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nWe start by proving (2), by explicitly constructing a resolution of \n$\\OO_{[Gx]}$. \n\nLet $\\varphi_x$ be a holomorphic section of $\\OO(1)$ which has a simple\nzero at $x$. Then we have a short exact sequence of sheaves (not\nequivariant):\n$$\n0\\to\\OO\\xxto{\\varphi_x}\\OO(1)\\to \\OO_{[x]}\\to 0\n$$\nTensoring it with $\\OO(-1)$, we get a sequence\n$$\n0\\to\\OO(-1)\\xxto{\\varphi_x}\\OO\\to \\OO_{[x]}\\to 0\n$$\nNow let us apply to this sequence functor $\\Ind$ defined by\n\\eqref{e:ind}. Using \\leref{l:frobenius}, we see that it gives a\n$\\Gbar$-equivariant short exact sequence \n$$\n0\\to R_1\\ttt \\OO(-1)\\to R_0\\ttt\\OO\\to \\OO_{[Gx]}\\to 0\n$$\nwhich gives equality $\\delta=\\delta_0-\\delta_1$, thus proving part (2) of the\ntheorem.\n\n\n\nPart (1) follows from (2).\n\nSince $\\OO_{[Gx]}\\ttt \\OO(-2)\\simeq \\OO_{[Gx]}$, we get $C\\delta=\\delta$.\nTo compute $C\\delta_0$, note that the same argument as in the proof of\npart (2) also gives a short exact sequence \n$$\n0\\to R_0\\ttt \\OO\\to R_1\\ttt\\OO(1)\\to \\OO_{[Gx]}\\to 0\n$$\nthus giving $C^{-1}\\delta_1=\\delta+\\delta_0=2\\delta+\\delta_1$. \n\nTo prove part (4), recall notation\n$\\<[X],[Y]\\>=\\dim\\Hom(X,Y)-\\dim\\Ext^1(X,Y)$. Then Serre duality\nimmediately gives \n$$\n\\=-\\.\n$$\nSince $C\\delta=\\delta$, we get \n$$\n(\\delta,x)=\\+\\<\\delta,x\\>=\\-\\=0\n$$\n\n\nPart (4) implies that $\\delta$ is an imaginary root. Moreover, since by\n\\coref{c:generators_of_K(C)} classes $[X_i]$, $[X_i(-1)]$ are free\ngenerators of $L$. Since some of $d_i$ are equal to 1, it follows\nfrom part (2) that for any $k>1$, $\\delta\/k\\notin L$; thus, $\\delta$ must be\na generator of the set of imaginary roots. \n\\end{proof}\n\nSince every indecomposable object in $\\D^b(\\C)$ is of the form\n$\\F[n]$, where $\\F$ is an indecomposable $G$-equivariant sheaf\n(\\thref{t:indecomposable}), it follows that every root $\\alpha\\in\\Delta$\ncan be written as either $\\alpha=[\\F]$ or $\\alpha=-[\\F]$, thus giving some\nsplitting of $\\Delta$ into positive and negative roots. This polarization\ncan be described explicitly. \n\nRecall from algebraic geometry (see, e.g., \\cite[Exercises II.6.10,\nII.6.12]{hartshorne}) that for any coherent sheaf, we can\ndefine two integer numbers, its {\\em rank} and {\\em degree}. In\nparticular, for a locally free sheaf $\\F=X\\ttt \\OO(n)$ (where $X$ is a\nfinite-dimensional vector space), we have \n\\begin{align*}\n&\\rk (X\\ttt \\OO(n))=\\dim X,\\\\\n&\\deg (X\\ttt \\OO(n))= n\\dim X.\n\\end{align*}\nDegree and rank give well-defined linear maps $K\\to \\Z$, where $K$\nis the Grothendieck group of the category of coherent sheaves. \n\nIn particular, we can define rank and degree for a $G$-equivariant\nsheaf, ignoring the equivariant structure; this gives linear maps\n$K(\\C)\\to\\Z$, which we also denote by $\\rk, \\deg$. \n\n\n\\begin{lemma}\\label{l:rank}\n\\par\\indent\n \\begin{enumerate}\n \\item If $\\F\\in\\C$ is a non-zero free sheaf, then $\\rk \\F>0$.\n \\item If $\\F\\in\\C$ is a non-zero torsion sheaf, then $\\rk \\F=0$,\n $\\deg\\F>0$. \n \\item For any $x\\in K(\\C)$, we have \n $$\n \\rk (x)=(x,\\delta_0)=(x,\\delta_1)\n $$\n where $\\delta_0$, $\\delta_1$ are defined in \\thref{t:imaginary_root}.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n The first two parts are well-known. \n \n To check $\\rk (x)=(x,\\delta_0)=(x,\\delta_1)$, it suffices to check it\n for $x=[X_i]$, $i\\in I_0$ and $x=[X_i(-1)], i\\in I_1$ (by\n \\exref{x:standard2}, they generate $L$). If $i\\in I_0$,\n then \n $$\n ([X_i], \\delta_0)=\\dim \\Hom_G(X_i,\\sum_{j\\in I_0} d_jX_j)=d_i=\\rk X_i\n $$\n thus proving $\\rk (x)=(x,\\delta_0)$. Since $\\delta_0-\\delta_1=\\delta$ is in the\n kernel of $(\\, , \\,)$ (\\thref{t:imaginary_root}), this implies\n $(x,\\delta_0)=(x,\\delta_1)$. \n \n If $i\\in I_1$, then \n $$\n ([X_i(-1)], \\delta_1)=\\dim \\Hom_G(X_i(-1),\\sum_{j\\in I_1}\n d_jX_j(-1))=d_i=\\rk X_i\n $$\n thus proving $\\rk (x)=(x,\\delta_1)$. Since $\\delta_0-\\delta_1=\\delta$ is in the\n kernel of $(\\, , \\,)$ (\\thref{t:imaginary_root}), this implies\n $(x,\\delta_0)=(x,\\delta_1)$.\n\n\\end{proof}\n\nNote that the functional $\\deg x$ can not be written in terms of the\nform $(\\, , \\,)$: indeed, $\\deg \\delta =|\\Gbar|=|G|\/2$, but $(\\delta,\n\\cdot)=0$. \n\n\\begin{theorem}\nLet $\\alpha\\in \\Delta$.\n\\begin{enumerate}\n \\item $\\alpha=[\\F]$ for some indecomposable free sheaf $\\F\\in \\C$ \n iff $\\rk(\\alpha)=(\\alpha,\\delta_0)>0$\n \\item $\\alpha=-[\\F]$ for some indecomposable free sheaf $\\F\\in \\C$ \n iff $\\rk(\\alpha)=(\\alpha,\\delta_0)<0$\n \\item $\\alpha=[\\F]$ for some indecomposable torsion sheaf $\\F\\in \\C$ \n iff $\\rk(\\alpha)=(\\alpha,\\delta_0)=0$, $\\deg(\\alpha)>0$\n \\item $\\alpha=-[\\F]$ for some indecomposable torsion sheaf $\\F\\in \\C$ \n iff $\\rk(\\alpha)=(\\alpha,\\delta_0)=0$, $\\deg(\\alpha)<0$\n\\end{enumerate}\n\\end{theorem}\nThus, we see that we have a triangular decomposition of $\\Delta$:\n\\begin{equation}\\label{e:polarization}\n \\begin{aligned}\n &\\Delta=\\Delta_+\\sqcup\\Delta_0\\sqcup \\Delta_-\\\\\n &\\Delta_+=\\{\\alpha\\in \\Delta\\st \\rk(\\alpha)>0\\}=\\{[\\F],\\ \\F\\text{ --- a free\n indecomposable sheaf}\\}\\\\\n &\\quad \\text{(note that $\\Delta_+$ is exactly the set of vertices of\n ${\\widehat{Q}}$)}\\\\\n &\\Delta_-=\\{\\alpha\\in \\Delta\\st \\rk(\\alpha)<0\\}=\\{-[\\F],\\ \\F\\text{ --- a free\n indecomposable sheaf}\\}\\\\\n &\\Delta_0=\\{\\alpha\\in \\Delta\\st \\rk(\\alpha)=0\\}=\\{\\pm[\\F],\\ \\F\\text{ --- a free\n torsion sheaf}\\}\n \\end{aligned}\n\\end{equation}\nThe set $\\Delta_+$ has been discussed by Schiffmann \\cite{schiffmann2}, who\nused notation ${\\widehat{Q}}_+$ and denoted the corresponding subalgebra in the\nloop algebra by $\\mathcal{L}\\mathfrak{n}$. Note, however, that this\nnotation is somewhat misleading: $\\mathcal{L}\\mathfrak{n}$ is not the\nloop algebra of a positive part of the finite dimensional algebra\n$\\bar\\g$, which easily follows from the fact that there are real roots in \n$\\Delta_0$ (see example in the next section).\n \n\\begin{theorem}\\label{t:C^g}\nLet $g=|\\Gbar|=|G|\/2$. Then for any $x\\in L$ we have \n\\begin{equation}\\label{e:C^g}\nC^g (x)=x-2(\\rk x)\\delta \n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nLet $\\varphi_x$ be a section of the sheaf $\\OO(1)$ which has a single zero at\ngeneric point $x$. Then we have a short exact sequence \n$$\n0\\to \\OO\\xxto{\\varphi_x} \\OO(1)\\to \\OO_{[x]}\\to 0\n$$\nwhich, however, is not equivariant even under the action of $\\{\\pm\nI\\}\\subset \\mathrm{SU(2)}$. To fix it we consider $\\varphi_x^2$ which gives the following\n$\\Z_2$-equivariant sequence\n$$\n0\\to \\OO(-2)\\xxto{\\varphi^2_x} \\OO\\to \\OO_{2[x]}\\to 0\n$$\nNow let us take product of pullbacks of $\\varphi_x^2$ under all $g\\in\\Gbar$\n$$\n0\\to \\OO(-2g)\\xxto{\\prod_{g\\in\\Gbar}g^*\\varphi^2_x} \\OO\\to \\OO_{2[Gx]}\\to 0\n$$\n\nTensoring it with any locally free sheaf $\\F$, we get \n$$\n0\\to \\F(-2g)\\xxto{\\prod_{g\\in\\Gbar}g^*\\varphi^2_x} \\F\\to\n\\OO_{2[Gx]}\\ttt \\F_x\\to 0\n$$\nwhich implies\n$C^g[\\F]-[\\F]+2(\\rk \\F)\\delta=0$.\n\\end{proof}\n\nThis result --- that $C^g$ is a translation --- was known before and\ncan be proved without the use of equivariant sheaves, see e.g.\nSteinberg \\cite{steinberg}. However, the approach via equivariant\nsheaves also provides a nice interpretation for the corresponding\nfunctional as the rank of the sheaf.\n\nComparing \\eqref{e:C^g} with the description of the action of Coxeter\nelement in the language of representations of the quiver, we see that rank\nis closely related to the notion of defect $\\partial_c(x)$ as defined in\n\\cite{drab-ringel}, namely \n$$\n\\rk(x)=-\\frac{1}{2}\\partial_c x\n$$\nTherefore, $\\Delta_0$ is exactly the set of indecomposable objects of defect\nzero, which shows that torsion sheaves correspond to regular\nrepresentations. \n\n\n\n\\begin{corollary}\n \\par\\noindent\n \\begin{enumerate}\n \\item For any $\\alpha\\in \\Delta_0$, $C^gx=x$; in particular, $C$-orbit\n of $\\alpha$ is finite.\n \\item $C$ acts freely on $\\Delta_+$, and the set of orbits $\\Delta_+\/C$\n is naturally in bijection with $Q$. \n \\end{enumerate}\n\\end{corollary}\n\n\n\\section{Example: $\\widehat{A}_n$}\nIn this section, we consider the example of the cyclic group of even\norder: $G=\\Z_n$, $n=2k$. \n\nThe irreducible representations of this group are $X_i$, $i\\in \\Z_n$;\nall of them are one-dimensional. The corresponding Dynkin diagram $Q$\nis the cycle of length $n$. \n\nThe root system $\\Delta(Q)$ can be described as follows. Let $V$ be a\nreal vector space of dimension $n+1$, with basis $\\delta, e_i$, $i\\in\n\\Z_n$. Define inner product in $V$ by $(e_i,e_j)=\\delta_{ij}$,\n$(v,\\delta)=0$. Then \n\\begin{equation}\n \\Delta(Q)=\\{e_i-e_j+a\\delta, i\\ne j\\in \\Z_n, a\\in \\frac{j-i}{n}+\\Z\\}\n\\end{equation}\nThe simple roots are \n$$\n\\alpha_i=e_i-e_{i+1}+\\frac{1}{n}\\delta,\\quad i\\in \\Z_n\n$$\nso that $\\sum_i\\alpha_i=\\delta$. The simple reflections $s_i$ are given by \n\\begin{align*}\n&s_i(e_i)=e_{i+1}-\\frac{1}{n}\\delta\\\\\n&s_i(e_{i+1})=e_{i}+\\frac{1}{n}\\delta\\\\\n&s_i(e_j)=e_j,\\quad j\\ne i,i+1\n\\end{align*}\n\n\nIt is easy to see that this description of\n$\\Delta(Q)$, while unusual, is equivalent to the standard\ndescription of the affine root system $\\widehat{A}_{n-1}$.\n\nWe choose standard height function $h$:\n\\begin{equation}\nh(i)=\\begin{cases}0,& i\\text{ even}\\\\\n 1, &i\\text{ odd}\n \\end{cases}\n\\end{equation}\nThe corresponding Coxeter element $C$ is \n$$\nC=\\left(\\prod_{i \\text{ odd}} s_i\\right)\n \\left(\\prod_{i \\text{ even}} s_i\\right)\n$$\nThe action of $C$ on $\\Delta(Q)$ is given by \n\\begin{equation*}\nC(e_i)=\\begin{cases}e_{i+2}-\\frac{2}{n}\\delta,& i\\text{ even}\\\\\n e_{i-2}+\\frac{2}{n}\\delta,& i\\text{ odd}\n \\end{cases}\n\\end{equation*}\nThus, we have \n\\begin{equation*}\nC^{n\/2}(e_i)=\\begin{cases}e_{i}-\\delta,& i\\text{ even}\\\\\n e_{i}+\\delta,& i\\text{ odd}\n \\end{cases}\n\\end{equation*}\nwhich implies\n\\begin{equation}\n\\begin{aligned}\n&C^{n\/2}(\\alpha)=\\alpha-(\\alpha,\\varepsilon)\\delta,\\\\\n&\\quad \\varepsilon = \\sum_{i \\text{ even}}e_i-\\sum_{i \\text{ odd}}e_i\\equiv \n\\sum_{i \\text{ even}}\\alpha_i\\equiv -\\sum_{i \\text{ odd}}\\alpha_i\\mod \\Z\\delta\n\\end{aligned}\n\\end{equation}\n(compare with \\thref{t:C^g}, \\leref{l:rank} ). \n\nExplicitly, we can write \n\\begin{align*}\n&C^{n\/2}(\\alpha)=\\begin{cases}\n \\alpha, &i\\equiv j\\mod 2 \\\\\n \\alpha-2\\delta, &(i,j)\\equiv (0,1)\\mod 2 \\\\\n \\alpha+2\\delta, &(i,j)\\equiv (1,0)\\mod 2 \n \\end{cases},\\\\\n&\\quad \\alpha=e_i-e_j+a\\delta\n\\end{align*}\nThus, in this case we have \n\\begin{align*}\n\\Delta_0=\\{e_i-e_j+a\\delta\\}, \\quad i\\equiv j\\mod 2\\\\\n\\Delta_+=\\{e_i-e_j+a\\delta\\}, \\quad i\\text{ even}, j\\text{ odd}\\\\\n\\Delta_-=\\{e_i-e_j+a\\delta\\}, \\quad i\\text{ odd}, j\\text{ even}\\\\\n\\end{align*}\n\n\n\n\\firef{f:An_graph} shows a segment of the AR graph ${\\widehat{Q}}$ for\nthis root system.\n\n\n\\begin{figure}[ht]\n$$ \n\\xy\n(0,0)*+{X_{i-2}}=\"A\"+(0,-3)*+{\\scriptstyle\n e_{i-2}-e_{i-1}+\\frac{1}{n}\\delta};\n(30,0)*+{X_{i}}=\"B\"+(0,-3)*{\\scriptstyle\n e_{i}-e_{i+1}+\\frac{1}{n}\\delta};\n(60,0)*+{X_{i+2}}=\"C\"+(0,-3)*{\\scriptstyle\n e_{i+2}-e_{i+3}+\\frac{1}{n}\\delta};\n(90,0)*+{X_{i+4}}=\"D\"+(0,-3)*{\\scriptstyle\n e_{i+4}-e_{i+5}+\\frac{1}{n}\\delta};\n(15,15)*+{X_{i-1}(1)}=\"E\"+(0,-3)*{\\scriptstyle\n e_{i-2}-e_{i+1}+\\frac{3}{n}\\delta}=\"EE\";\n(45,15)*+{X_{i+1}(1)}=\"F\"+(0,-3)*{\\scriptstyle\n e_{i}-e_{i+3}+\\frac{3}{n}\\delta}=\"FF\";\n(75,15)*+{X_{i+3}(1)}=\"G\"+(0,-3)*{\\scriptstyle\n e_{i+2}-e_{i+5}+\\frac{3}{n}\\delta}=\"GG\";\n(0,30)*+{X_{i-2}(2)}=\"H\"+(0,-3)*+{\\scriptstyle\n e_{i-4}-e_{i+1}+\\frac{5}{n}\\delta}=\"HH\";\n(30,30)*+{X_{i}(2)}=\"I\"+(0,-3)*{\\scriptstyle\n e_{i-2}-e_{i+3}+\\frac{5}{n}\\delta}=\"II\";\n(60,30)*+{X_{i+2}(2)}=\"J\"+(0,-3)*{\\scriptstyle\n e_{i}-e_{i+5}+\\frac{5}{n}\\delta}=\"JJ\";\n(90,30)*+{X_{i+4}(2)}=\"K\"+(0,-3)*{\\scriptstyle\n e_{i+2}-e_{i+7}+\\frac{5}{n}\\delta}=\"KK\";\n{\\ar \"A\";\"EE\"};\n{\\ar \"B\";\"EE\"};\n{\\ar \"B\";\"FF\"};\n{\\ar \"C\";\"FF\"};\n{\\ar \"C\";\"GG\"};\n{\\ar \"D\";\"GG\"};\n{\\ar \"E\";\"HH\"};\n{\\ar \"E\";\"II\"};\n{\\ar \"F\";\"II\"};\n{\\ar \"F\";\"JJ\"};\n{\\ar \"G\";\"JJ\"};\n{\\ar \"G\";\"KK\"};\n\\endxy\n$$\n \\caption{Fragment of graph ${\\widehat{Q}}$ for root system of type $A$.\n Here $i$ is an even number. The figure also shows, for each $X_q\\in\n {\\widehat{Q}}$, the image of $R\\Ph_h[X_q]$ for the standard choice of height\n function $h$ as in \\exref{x:standard}.} \\label{f:An_graph}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}