In the present paper, we prove the retract rationality of the classifying spaces $BG$ for several types of finite connected group schemes $G$ over algebraically closed fields $k$ of positive characteristic $p>0$. In particular, we prove the retract rationality for the finite simple group schemes $G$ associated with the generalized Witt algebras in specific cases. To this end, we study the automorphism group schemes of the generalized Witt algebras and establish triangulations for them. Moreover, we extend the notion of Witt--Ree algebra to general base rings and discuss their properties.
{"title":"On retract rationality for finite connected group schemes","authors":"Shusuke Otabe","doi":"arxiv-2409.08604","DOIUrl":"https://doi.org/arxiv-2409.08604","url":null,"abstract":"In the present paper, we prove the retract rationality of the classifying\u0000spaces $BG$ for several types of finite connected group schemes $G$ over\u0000algebraically closed fields $k$ of positive characteristic $p>0$. In\u0000particular, we prove the retract rationality for the finite simple group\u0000schemes $G$ associated with the generalized Witt algebras in specific cases. To\u0000this end, we study the automorphism group schemes of the generalized Witt\u0000algebras and establish triangulations for them. Moreover, we extend the notion\u0000of Witt--Ree algebra to general base rings and discuss their properties.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253736","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 $Lambda$ an artin algebra and $n geq 2$. We study how to compute the left and right degrees of irreducible morphisms between complexes in a generalized standard Auslander-Reiten component of ${mathbf{C_n}({rm proj}, Lambda)}$ with length. We give conditions under which the kernel and the cokernel of irreducible morphisms between complexes in $mathbf{C_n}({rm proj}, Lambda)$ belong to such a category. For a finite dimensional hereditary algebra $H$ over an algebraically closed field, we determine when an irreducible morphism has finite left (or right) degree and we give a characterization, depending on the degrees of certain irreducible morphisms, under which $mathbf{C_n}({rm proj} ,H)$ is of finite type.
{"title":"On the degree in categories of complexes of fixed size","authors":"Claudia Chaio, Isabel Pratti, Maria Jose Souto","doi":"arxiv-2409.08758","DOIUrl":"https://doi.org/arxiv-2409.08758","url":null,"abstract":"We consider $Lambda$ an artin algebra and $n geq 2$. We study how to\u0000compute the left and right degrees of irreducible morphisms between complexes\u0000in a generalized standard Auslander-Reiten component of ${mathbf{C_n}({rm\u0000proj}, Lambda)}$ with length. We give conditions under which the kernel and\u0000the cokernel of irreducible morphisms between complexes in $mathbf{C_n}({rm\u0000proj}, Lambda)$ belong to such a category. For a finite dimensional\u0000hereditary algebra $H$ over an algebraically closed field, we determine when an\u0000irreducible morphism has finite left (or right) degree and we give a\u0000characterization, depending on the degrees of certain irreducible morphisms,\u0000under which $mathbf{C_n}({rm proj} ,H)$ is of finite type.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253734","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 $A={rm mathbb{k}}Q/mathcal{I}$ be a gentle algebra. We provide a bijection between non-projective indecomposable Gorenstein projective modules over $A$ and special recollements induced by an arrow $a$ on any full-relational oriented cycle $mathscr{C}$, which satisfies some interesting properties, for example, the tensor functor $-otimes_A A/Avarepsilon A$ sends Gorenstein projective module $aA$ to an indecomposable projective $A/Avarepsilon A$-module; and $-otimes_A A/Avarepsilon A$ preserves Gorenstein projective objects if any two full-relational oriented cycles do not have common vertex.
{"title":"Recollements and Gorenstein projective modules for gentle algebras","authors":"Yu-Zhe Liu, Dajun Liu, Xin Ma","doi":"arxiv-2409.08686","DOIUrl":"https://doi.org/arxiv-2409.08686","url":null,"abstract":"Let $A={rm mathbb{k}}Q/mathcal{I}$ be a gentle algebra. We provide a\u0000bijection between non-projective indecomposable Gorenstein projective modules\u0000over $A$ and special recollements induced by an arrow $a$ on any\u0000full-relational oriented cycle $mathscr{C}$, which satisfies some interesting\u0000properties, for example, the tensor functor $-otimes_A A/Avarepsilon A$ sends\u0000Gorenstein projective module $aA$ to an indecomposable projective\u0000$A/Avarepsilon A$-module; and $-otimes_A A/Avarepsilon A$ preserves\u0000Gorenstein projective objects if any two full-relational oriented cycles do not\u0000have common vertex.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253733","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}
Terwilliger algebras are a subalgebra of a matrix algebra that are constructed from association schemes over finite sets. In 2010, Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative. In that paper she gave five equivalent conditions for a Terwilliger algebra to be almost commutative. In this paper, we provide a classification of which groups result in an almost commutative Terwilliger algebra when looking at the group association scheme (the Schur ring generated by the conjugacy classes of the group). In particular, we show that all such groups are either abelian, or Camina groups. Following this classification, we then compute the dimension and non-primary primitive idempotents for each Terwilliger algebra of this form for the first three types of groups whose group association scheme gives an almost commutative Terwilliger algebra. The final case will be considered in a second paper.
{"title":"Almost Commutative Terwilliger Algebras of Group Association Schemes I: Classification","authors":"Nicholas L. Bastian, Stephen P. Humphries","doi":"arxiv-2409.09167","DOIUrl":"https://doi.org/arxiv-2409.09167","url":null,"abstract":"Terwilliger algebras are a subalgebra of a matrix algebra that are\u0000constructed from association schemes over finite sets. In 2010, Rie Tanaka\u0000defined what it means for a Terwilliger algebra to be almost commutative. In\u0000that paper she gave five equivalent conditions for a Terwilliger algebra to be\u0000almost commutative. In this paper, we provide a classification of which groups\u0000result in an almost commutative Terwilliger algebra when looking at the group\u0000association scheme (the Schur ring generated by the conjugacy classes of the\u0000group). In particular, we show that all such groups are either abelian, or\u0000Camina groups. Following this classification, we then compute the dimension and\u0000non-primary primitive idempotents for each Terwilliger algebra of this form for\u0000the first three types of groups whose group association scheme gives an almost\u0000commutative Terwilliger algebra. The final case will be considered in a second\u0000paper.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"201 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253727","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}
Gentle algebras are a class of special biserial algebra whose representation theory has been thoroughly described. In this paper, we consider the infinite dimensional generalizations of gentle algebras, referred to as locally gentle algebras. We give combinatorial descriptions of the center, spectrum, and homological dimensions of a locally gentle algebra, including an explicit injective resolution. We classify when these algebras are Artin-Schelter Gorenstein, Artin-Schelter regular, and Cohen-Macaulay, and provide an analogue of Stanley's theorem for locally gentle algebras.
{"title":"Homological conditions on locally gentle algebras","authors":"S. Ford, A. Oswald, J. J. Zhang","doi":"arxiv-2409.08333","DOIUrl":"https://doi.org/arxiv-2409.08333","url":null,"abstract":"Gentle algebras are a class of special biserial algebra whose representation\u0000theory has been thoroughly described. In this paper, we consider the infinite\u0000dimensional generalizations of gentle algebras, referred to as locally gentle\u0000algebras. We give combinatorial descriptions of the center, spectrum, and\u0000homological dimensions of a locally gentle algebra, including an explicit\u0000injective resolution. We classify when these algebras are Artin-Schelter\u0000Gorenstein, Artin-Schelter regular, and Cohen-Macaulay, and provide an analogue\u0000of Stanley's theorem for locally gentle algebras.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"201 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253735","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 $G$ be a simple algebraic group over an algebraically closed field $Bbbk$ of positive characteristic. We consider the questions of when the tensor product of two simple $G$-modules is multiplicity free or completely reducible. We develop tools for answering these questions in general, and we use them to provide complete answers for the groups $G = mathrm{SL}_3(Bbbk)$ and $G = mathrm{Sp}_4(Bbbk)$.
{"title":"Multiplicity free and completely reducible tensor products for $mathrm{SL}_3(Bbbk)$ and $mathrm{Sp}_4(Bbbk)$","authors":"Jonathan Gruber, Gaëtan Mancini","doi":"arxiv-2409.07888","DOIUrl":"https://doi.org/arxiv-2409.07888","url":null,"abstract":"Let $G$ be a simple algebraic group over an algebraically closed field\u0000$Bbbk$ of positive characteristic. We consider the questions of when the\u0000tensor product of two simple $G$-modules is multiplicity free or completely\u0000reducible. We develop tools for answering these questions in general, and we\u0000use them to provide complete answers for the groups $G = mathrm{SL}_3(Bbbk)$\u0000and $G = mathrm{Sp}_4(Bbbk)$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"275 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187291","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 article we give an overview of the Plancherel theory for Riemannian symmetric spaces Z = G/K. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel theorem for Z can be proven from these methods.
本文概述了黎曼对称空间 Z = G/K 的 Plancherel 理论。特别是，我们通过对黎曼对称空间的解释，说明了最近开发的用于实球面空间的 Plancherel 理论方法，并解释了如何通过这些方法证明 Z 的哈里什-钱德拉 Plancherel 定理。
{"title":"On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces","authors":"Bernhard Krötz, Job J. Kuit, Henrik Schlichtkrull","doi":"arxiv-2409.08113","DOIUrl":"https://doi.org/arxiv-2409.08113","url":null,"abstract":"In this article we give an overview of the Plancherel theory for Riemannian\u0000symmetric spaces Z = G/K. In particular we illustrate recently developed\u0000methods in Plancherel theory for real spherical spaces by explicating them for\u0000Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel\u0000theorem for Z can be proven from these methods.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187289","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 the theory of geometric Eisenstein series and constant term functors for $ell$-adic sheaves on stacks of bundles on the Fargues-Fontaine curve. In particular, we prove essentially optimal finiteness theorems for these functors, analogous to the usual finiteness properties of parabolic inductions and Jacquet modules. We also prove a geometric form of Bernstein's second adjointness theorem, generalizing the classical result and its recent extension to more general coefficient rings proved in [Dat-Helm-Kurinczuk-Moss]. As applications, we decompose the category of sheaves on $mathrm{Bun}_G$ into cuspidal and Eisenstein parts, and show that the gluing functors between strata of $mathrm{Bun}_G$ are continuous in a very strong sense.
我们发展了几何爱森斯坦级数和 $ell$-adic sheaves on stacks of bundles on the Fargues-Fontainecurve 的常数项函数理论。特别是，我们证明了这些函数本质上的最优有限性定理，类似于抛物线引入和雅克特模块的通常有限性性质。我们还证明了伯恩斯坦第二邻接性定理的几何形式，推广了[Dat-Helm-Kurinczuk-Moss]中证明的经典结果及其对更一般系数环的再推广。作为应用，我们把$mathrm{Bun}_G$上的舍弗类分解成簕杜鹃部分和爱森斯坦部分，并证明了$mathrm{Bun}_G$的层之间的胶合函数在非常强的意义上是连续的。
{"title":"Geometric Eisenstein series I: finiteness theorems","authors":"Linus Hamann, David Hansen, Peter Scholze","doi":"arxiv-2409.07363","DOIUrl":"https://doi.org/arxiv-2409.07363","url":null,"abstract":"We develop the theory of geometric Eisenstein series and constant term\u0000functors for $ell$-adic sheaves on stacks of bundles on the Fargues-Fontaine\u0000curve. In particular, we prove essentially optimal finiteness theorems for\u0000these functors, analogous to the usual finiteness properties of parabolic\u0000inductions and Jacquet modules. We also prove a geometric form of Bernstein's\u0000second adjointness theorem, generalizing the classical result and its recent\u0000extension to more general coefficient rings proved in\u0000[Dat-Helm-Kurinczuk-Moss]. As applications, we decompose the category of\u0000sheaves on $mathrm{Bun}_G$ into cuspidal and Eisenstein parts, and show that\u0000the gluing functors between strata of $mathrm{Bun}_G$ are continuous in a very\u0000strong sense.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187302","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 $pge 5$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite symmetric group. To every irreducible character of $P$ we associate a collection of labelled, complete $p$-ary trees. The main results of this article describe Sylow branching coefficients for symmetric groups for all irreducible characters of $P$ in terms of some combinatorial properties of these trees, extending previous work on the linear characters of $P$.
{"title":"Sylow branching trees","authors":"Eugenio Giannelli, Stacey Law","doi":"arxiv-2409.07575","DOIUrl":"https://doi.org/arxiv-2409.07575","url":null,"abstract":"Let $pge 5$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite\u0000symmetric group. To every irreducible character of $P$ we associate a\u0000collection of labelled, complete $p$-ary trees. The main results of this\u0000article describe Sylow branching coefficients for symmetric groups for all\u0000irreducible characters of $P$ in terms of some combinatorial properties of\u0000these trees, extending previous work on the linear characters of $P$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187290","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}
For $ngeq 1$, we construct the Hilbert scheme of $n$ points on any crepant partial resolution of a Kleinian singularity as a Nakajima quiver variety for an explicit GIT stability parameter. We provide both a short proof involving a combinatorial argument, in which the isomorphism is implicit, and a more satisfying geometric proof where the isomorphic is constructed explicitly. As a corollary, we compute the nef and movable cones of the Hilbert scheme of $n$ points on any crepant partial resolution of a Kleinian singularity in terms of the summands of a tilting bundle on the surface.
{"title":"Hilbert schemes for crepant partial resolutions","authors":"Alastair Craw, Ruth Pugh","doi":"arxiv-2409.07408","DOIUrl":"https://doi.org/arxiv-2409.07408","url":null,"abstract":"For $ngeq 1$, we construct the Hilbert scheme of $n$ points on any crepant\u0000partial resolution of a Kleinian singularity as a Nakajima quiver variety for\u0000an explicit GIT stability parameter. We provide both a short proof involving a\u0000combinatorial argument, in which the isomorphism is implicit, and a more\u0000satisfying geometric proof where the isomorphic is constructed explicitly. As a\u0000corollary, we compute the nef and movable cones of the Hilbert scheme of $n$\u0000points on any crepant partial resolution of a Kleinian singularity in terms of\u0000the summands of a tilting bundle on the surface.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"164 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187301","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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