In a previous paper, we introduced quasi-BPS categories for moduli stacks of semistable Higgs bundles. Under a certain condition on the rank, Euler characteristic, and weight, the quasi-BPS categories (called BPS in this case) are non-commutative analogues of Hitchin integrable systems. We proposed a conjectural equivalence between BPS categories which swaps Euler characteristics and weights. The conjecture is inspired by the Dolbeault Geometric Langlands equivalence of Donagi--Pantev, by the Hausel--Thaddeus mirror symmetry, and by the $chi$-independence phenomenon for BPS invariants of curves on Calabi-Yau threefolds. In this paper, we show that the above conjecture holds at the level of topological K-theories. When the rank and the Euler characteristic are coprime, such an isomorphism was proved by Groechenig--Shen. Along the way, we show that the topological K-theory of BPS categories is isomorphic to the BPS cohomology of the moduli of semistable Higgs bundles.
在上一篇论文中，我们介绍了可迷惑希格斯束的模叠的准BPS范畴。在秩，欧拉特性和权重的特定条件下，准 BPS 范畴（这里称为 BPS）是希金可积分系统的非交换类似物。我们提出了 BPS 范畴之间的等价猜想，即交换欧拉特征和权重。这一猜想受到了多纳吉--潘特夫（Donagi--Pantev）的多尔博几何朗兰兹等价、豪塞尔--塔德斯镜像对称性以及卡拉比--尤三折上曲线的BPS不变量的$chi$-independence现象的启发。在本文中，我们证明了上述猜想在拓扑 K 理论层面上成立。当秩和欧拉特征为共素时，这种同构由格罗切尼-申证明。同时，我们还证明了BPS范畴的拓扑K理论与半稳希格斯束模态的BPS同调同构。
{"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}
Using the language of Seshadri stratifications we develop a geometrical interpretation of Lakshmibai-Seshadri-tableaux and their associated standard monomial bases. These tableaux are a generalization of Young-tableaux and De-Concini-tableaux to all Dynkin types. More precisely, we construct filtrations of multihomogeneous coordinate rings of Schubert varieties, such that the subquotients are one-dimensional and indexed by standard tableaux.
利用塞沙德里分层语言，我们对拉克希米拜-塞沙德里台面及其相关的标准单数基进行了几何解释。这些台构是 Young 台构和 De-Concini 台构对所有 Dynkin 类型的概括。更确切地说，我们构造了舒伯特变项多同质坐标环的过滤，使得子项是一维的，并以标准表项为索引。
{"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}
We consider one-parameter families of quadratic-phase integral transforms which generalize the fractional Fourier transform. Under suitable regularity assumptions, we characterize the one-parameter groups formed by such transforms. Necessary and sufficient conditions for continuous dependence on the parameter are obtained in L2, pointwise, and almost-everywhere senses.
{"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}
In previous work, we associated a module $T(i)$ to every segment $i$ of a link diagram $K$ and showed that there is a poset isomorphism between the submodules of $T(i)$ and the Kauffman states of $K$ relative to $i$. In this paper, we show that the posets are distributive lattices and give explicit descriptions of the join irreducibles in both posets. We also prove that the subposet of join irreducible Kauffman states is isomorphic to the poset of the coefficient quiver of $T(i)$.
{"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}
We develop a functorial approach to the study of $n$-abelian categories by reformulating their axioms in terms of their categories of finitely presented functors. Such an approach allows the use of classical homological algebra and representation theory techniques to understand higher homological algebra. As an application, we present two possible generalizations of the axioms "every monomorphism is a kernel" and "every epimorphism is a cokernel" of an abelian category to $n$-abelian categories. We also specialize our results to modules over rings, thereby describing when the category of finitely generated projective modules over a ring is $n$-abelian. Moreover, we establish a correspondence for $n$-abelian categories with additive generators, which extends the higher Auslander correspondence.
{"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}
This article investigates Dehornoy's monomial representations for structure groups and Coxeter-like groups of a set-theoretic solution to the Yang-Baxter equation. Using the brace structure of these two groups and the language of cycle sets, we relate the irreducibility of monomial representations to the indecomposability of the solutions. Furthermore, in the case of an indecomposable solution, we show how to obtain these representations by induction from explicit one-dimensional representations.
{"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}
This paper is devoted to the study of the shifted Yangian $Y_n(sigma)$ associated to the general linear Lie algebra $mathfrak{gl}_n$ over a field of positive characteristic. We obtain an explicit description of the center $Z(Y_n(sigma))$ of $Y_n(sigma)$ in terms of parabolic generators, showing that it is generated by its Harish-Chandra center and its $p$-center.
{"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}
In this paper we construct an algebraic invariant attached to Galois representations over number fields. This invariant, which we call an Artin symmetric function, lives in a certain ring we introduce called the ring of arithmetic symmetric functions. This ring is built from a family of symmetric functions rings indexed by prime ideals of the base field. We prove many necessary basic results for the ring of arithmetic symmetric functions as well as introduce the analogues of some standard number-theoretic objects in this setting. We prove that the Artin symmetric functions satisfy the same algebraic properties that the Artin L-functions do with respect to induction, inflation, and direct summation of representations. The expansion coefficients of these symmetric functions in different natural bases are shown to be character values of representations of a compact group related to the original Galois group. In the most interesting case, the expansion coefficients into a specialized Hall-Littlewood basis come from new representations built from the original Galois representation using polynomial functors corresponding to modified Hall-Littlewood polynomials. Using a special case of the Satake isomorphism in type GL, as formulated by Macdonald, we show that the Artin symmetric functions yield families of functions in the (finite) global spherical Hecke algebras in type GL which exhibit natural stability properties. We compute the Mellin transforms of these functions and relate them to infinite products of shifted Artin L-functions. We then prove some analytic properties of these Dirichlet series and give an explicit expansion of these series using the Hall-Littlewood polynomial functors.
在本文中，我们构建了一个附加于数域上伽罗瓦表示的代数不变量。这个不变量被我们称为阿尔廷对称函数，它存在于我们引入的某个环中，这个环被称为算术对称函数环。这个环由基域素理想索引的对称函数环族构建而成。我们证明了算术对称函数环的许多必要的基本结果，并介绍了一些标准数论对象在这一集合中的类似物。我们证明了阿尔丁对称函数在归纳、膨胀和直接求和表示方面满足与阿尔丁 L 函数相同的代数特性。在不同的自然基中，对称函数的膨胀系数被证明是与原始伽罗瓦群相关的紧凑群的表征的特征值。在最有趣的情况下，在专门的霍尔-利特尔伍德基（Hall-Littlewood basis）中的展开系数来自使用与修正的霍尔-利特尔伍德多项式相对应的多项式函数从原始伽罗瓦表示建立的新表示。利用麦克唐纳（Macdonald）提出的类型 GL 中 Satake 同构的一个特例，我们证明了 Artin 对称函数在类型 GL 的（有限）全局球面 Hecke 代数中产生了函数族，这些函数族表现出天然的稳定性。我们计算了这些函数的梅林特变换，并将它们与移位阿尔丁 L 函数的无限乘积联系起来。然后，我们证明了这些 Dirichlets 系列的一些解析性质，并利用霍尔-利特尔伍德波伦函数给出了这些系列的显式展开。
{"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}
Let $V$ be a freely generated pregraded vertex superalgebra, $H$ a Hamiltonian operator of $V$, and $g$ a diagonalizable automorphism of V commuting with $H$ with modulus $1$ eigenvalues. We prove that the $(g, H)$-twisted Zhu algebra of $V$ has a PBW basis, is isomorphic to the universal enveloping algebra of some non-linear Lie superalgebra, and satisfies the commutativity of BRST cohomology functors, which generalizes results of De Sole and Kac. As applications, we compute the twisted Zhu algebras of affine vertex superalgebras and affine $W$-algebras.
{"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}
This is an expanded version of the notes by the second author of the lectures on Hitchin systems and their quantization given by the first author at the Beijing Summer Workshop in Mathematics and Mathematical Physics ``Integrable Systems and Algebraic Geometry" (BIMSA-2024).
{"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}
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