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Topological K-theory of quasi-BPS categories for Higgs bundles 希格斯束准 BPS 类别的拓扑 K 理论
arXiv - MATH - Representation Theory Pub Date : 2024-09-17 DOI: arxiv-2409.10800
Tudor Pădurariu, Yukinobu Toda
{"title":"Topological K-theory of quasi-BPS categories for Higgs bundles","authors":"Tudor Pădurariu, Yukinobu Toda","doi":"arxiv-2409.10800","DOIUrl":"https://doi.org/arxiv-2409.10800","url":null,"abstract":"In a previous paper, we introduced quasi-BPS categories for moduli stacks of\u0000semistable Higgs bundles. Under a certain condition on the rank, Euler\u0000characteristic, and weight, the quasi-BPS categories (called BPS in this case)\u0000are non-commutative analogues of Hitchin integrable systems. We proposed a\u0000conjectural equivalence between BPS categories which swaps Euler\u0000characteristics and weights. The conjecture is inspired by the Dolbeault\u0000Geometric Langlands equivalence of Donagi--Pantev, by the Hausel--Thaddeus\u0000mirror symmetry, and by the $chi$-independence phenomenon for BPS invariants\u0000of curves on Calabi-Yau threefolds. In this paper, we show that the above conjecture holds at the level of\u0000topological K-theories. When the rank and the Euler characteristic are coprime,\u0000such an isomorphism was proved by Groechenig--Shen. Along the way, we show that\u0000the topological K-theory of BPS categories is isomorphic to the BPS cohomology\u0000of the moduli of semistable Higgs bundles.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multiprojective Seshadri stratifications for Schubert varieties and standard monomial theory 舒伯特变体的多投影塞沙德里分层和标准单项式理论
arXiv - MATH - Representation Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11488
Henrik Müller
{"title":"Multiprojective Seshadri stratifications for Schubert varieties and standard monomial theory","authors":"Henrik Müller","doi":"arxiv-2409.11488","DOIUrl":"https://doi.org/arxiv-2409.11488","url":null,"abstract":"Using the language of Seshadri stratifications we develop a geometrical\u0000interpretation of Lakshmibai-Seshadri-tableaux and their associated standard\u0000monomial bases. These tableaux are a generalization of Young-tableaux and\u0000De-Concini-tableaux to all Dynkin types. More precisely, we construct\u0000filtrations of multihomogeneous coordinate rings of Schubert varieties, such\u0000that the subquotients are one-dimensional and indexed by standard tableaux.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"194 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalizations of the fractional Fourier transform and their analytic properties 分数傅里叶变换的一般化及其分析特性
arXiv - MATH - Representation Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11201
Yue Zhou
{"title":"Generalizations of the fractional Fourier transform and their analytic properties","authors":"Yue Zhou","doi":"arxiv-2409.11201","DOIUrl":"https://doi.org/arxiv-2409.11201","url":null,"abstract":"We consider one-parameter families of quadratic-phase integral transforms\u0000which generalize the fractional Fourier transform. Under suitable regularity\u0000assumptions, we characterize the one-parameter groups formed by such\u0000transforms. Necessary and sufficient conditions for continuous dependence on\u0000the parameter are obtained in L2, pointwise, and almost-everywhere senses.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Knot theory and cluster algebra III: Posets 结理论和聚类代数 III：Posets
arXiv - MATH - Representation Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11287
Véronique Bazier-Matte, Ralf Schiffler
{"title":"Knot theory and cluster algebra III: Posets","authors":"Véronique Bazier-Matte, Ralf Schiffler","doi":"arxiv-2409.11287","DOIUrl":"https://doi.org/arxiv-2409.11287","url":null,"abstract":"In previous work, we associated a module $T(i)$ to every segment $i$ of a\u0000link diagram $K$ and showed that there is a poset isomorphism between the\u0000submodules of $T(i)$ and the Kauffman states of $K$ relative to $i$. In this\u0000paper, we show that the posets are distributive lattices and give explicit\u0000descriptions of the join irreducibles in both posets. We also prove that the\u0000subposet of join irreducible Kauffman states is isomorphic to the poset of the\u0000coefficient quiver of $T(i)$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A functorial approach to $n$-abelian categories n$阿贝尔范畴的函数式方法
arXiv - MATH - Representation Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10438
Vitor Gulisz
{"title":"A functorial approach to $n$-abelian categories","authors":"Vitor Gulisz","doi":"arxiv-2409.10438","DOIUrl":"https://doi.org/arxiv-2409.10438","url":null,"abstract":"We develop a functorial approach to the study of $n$-abelian categories by\u0000reformulating their axioms in terms of their categories of finitely presented\u0000functors. Such an approach allows the use of classical homological algebra and\u0000representation theory techniques to understand higher homological algebra. As\u0000an application, we present two possible generalizations of the axioms \"every\u0000monomorphism is a kernel\" and \"every epimorphism is a cokernel\" of an abelian\u0000category to $n$-abelian categories. We also specialize our results to modules\u0000over rings, thereby describing when the category of finitely generated\u0000projective modules over a ring is $n$-abelian. Moreover, we establish a\u0000correspondence for $n$-abelian categories with additive generators, which\u0000extends the higher Auslander correspondence.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"40 4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Indecomposability and irreducibility of monomial representations for set-theoretical solutions to the Yang-Baxter equation 杨-巴克斯特方程集合论解的单项式表示的不可分性和不可还原性
arXiv - MATH - Representation Theory Pub Date : 2024-09-16 DOI: arxiv-2409.10648
Carsten Dietzel, Edouard Feingesicht, Silvia Properzi
{"title":"Indecomposability and irreducibility of monomial representations for set-theoretical solutions to the Yang-Baxter equation","authors":"Carsten Dietzel, Edouard Feingesicht, Silvia Properzi","doi":"arxiv-2409.10648","DOIUrl":"https://doi.org/arxiv-2409.10648","url":null,"abstract":"This article investigates Dehornoy's monomial representations for structure\u0000groups and Coxeter-like groups of a set-theoretic solution to the Yang-Baxter\u0000equation. Using the brace structure of these two groups and the language of cycle sets,\u0000we relate the irreducibility of monomial representations to the\u0000indecomposability of the solutions. Furthermore, in the case of an\u0000indecomposable solution, we show how to obtain these representations by\u0000induction from explicit one-dimensional representations.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The center of modular shifted Yangians and parabolic generators 模数转换杨氏中心和抛物线发电机
arXiv - MATH - Representation Theory Pub Date : 2024-09-15 DOI: arxiv-2409.09773
Hao Chang, Hongmei Hu
{"title":"The center of modular shifted Yangians and parabolic generators","authors":"Hao Chang, Hongmei Hu","doi":"arxiv-2409.09773","DOIUrl":"https://doi.org/arxiv-2409.09773","url":null,"abstract":"This paper is devoted to the study of the shifted Yangian $Y_n(sigma)$\u0000associated to the general linear Lie algebra $mathfrak{gl}_n$ over a field of\u0000positive characteristic. We obtain an explicit description of the center\u0000$Z(Y_n(sigma))$ of $Y_n(sigma)$ in terms of parabolic generators, showing\u0000that it is generated by its Harish-Chandra center and its $p$-center.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Artin Symmetric Functions 阿尔廷对称函数
arXiv - MATH - Representation Theory Pub Date : 2024-09-15 DOI: arxiv-2409.09643
Milo Bechtloff Weising
{"title":"Artin Symmetric Functions","authors":"Milo Bechtloff Weising","doi":"arxiv-2409.09643","DOIUrl":"https://doi.org/arxiv-2409.09643","url":null,"abstract":"In this paper we construct an algebraic invariant attached to Galois\u0000representations over number fields. This invariant, which we call an Artin\u0000symmetric function, lives in a certain ring we introduce called the ring of\u0000arithmetic symmetric functions. This ring is built from a family of symmetric\u0000functions rings indexed by prime ideals of the base field. We prove many\u0000necessary basic results for the ring of arithmetic symmetric functions as well\u0000as introduce the analogues of some standard number-theoretic objects in this\u0000setting. We prove that the Artin symmetric functions satisfy the same algebraic\u0000properties that the Artin L-functions do with respect to induction, inflation,\u0000and direct summation of representations. The expansion coefficients of these\u0000symmetric functions in different natural bases are shown to be character values\u0000of representations of a compact group related to the original Galois group. In\u0000the most interesting case, the expansion coefficients into a specialized\u0000Hall-Littlewood basis come from new representations built from the original\u0000Galois representation using polynomial functors corresponding to modified\u0000Hall-Littlewood polynomials. Using a special case of the Satake isomorphism in\u0000type GL, as formulated by Macdonald, we show that the Artin symmetric functions\u0000yield families of functions in the (finite) global spherical Hecke algebras in\u0000type GL which exhibit natural stability properties. We compute the Mellin\u0000transforms of these functions and relate them to infinite products of shifted\u0000Artin L-functions. We then prove some analytic properties of these Dirichlet\u0000series and give an explicit expansion of these series using the Hall-Littlewood\u0000polynomial functors.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Twisted Zhu algebras 扭曲朱代数
arXiv - MATH - Representation Theory Pub Date : 2024-09-15 DOI: arxiv-2409.09656
Naoki Genra
{"title":"Twisted Zhu algebras","authors":"Naoki Genra","doi":"arxiv-2409.09656","DOIUrl":"https://doi.org/arxiv-2409.09656","url":null,"abstract":"Let $V$ be a freely generated pregraded vertex superalgebra, $H$ a\u0000Hamiltonian operator of $V$, and $g$ a diagonalizable automorphism of V\u0000commuting with $H$ with modulus $1$ eigenvalues. We prove that the $(g,\u0000H)$-twisted Zhu algebra of $V$ has a PBW basis, is isomorphic to the universal\u0000enveloping algebra of some non-linear Lie superalgebra, and satisfies the\u0000commutativity of BRST cohomology functors, which generalizes results of De Sole\u0000and Kac. As applications, we compute the twisted Zhu algebras of affine vertex\u0000superalgebras and affine $W$-algebras.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hitchin systems and their quantization 希钦系统及其量化
arXiv - MATH - Representation Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09505
Pavel Etingof, Henry Liu
{"title":"Hitchin systems and their quantization","authors":"Pavel Etingof, Henry Liu","doi":"arxiv-2409.09505","DOIUrl":"https://doi.org/arxiv-2409.09505","url":null,"abstract":"This is an expanded version of the notes by the second author of the lectures\u0000on Hitchin systems and their quantization given by the first author at the\u0000Beijing Summer Workshop in Mathematics and Mathematical Physics ``Integrable\u0000Systems and Algebraic Geometry\" (BIMSA-2024).","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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