{"title":"The self-dual indecomposable modules in blocks with cyclic defect groups","authors":"Caroline Lassueur, John Murray","doi":"arxiv-2409.05562","DOIUrl":"https://doi.org/arxiv-2409.05562","url":null,"abstract":"Let $p$ be an odd prime and let $mathbf{B}$ be a $p$-block of a finite\u0000group, such that $mathbf{B}$ has cyclic defect groups. We describe the\u0000self-dual indecomposable $mathbf{B}$-modules and for each such module\u0000determine whether it is symplectic or orthogonal.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187122","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}
Alexander Bertoloni Meli, Teruhisa Koshikawa, Jonathan Leake
{"title":"Inequalities characterizing distinguished unipotent orbits","authors":"Alexander Bertoloni Meli, Teruhisa Koshikawa, Jonathan Leake","doi":"arxiv-2409.06006","DOIUrl":"https://doi.org/arxiv-2409.06006","url":null,"abstract":"In this paper we prove a new characterization of the distinguished unipotent\u0000orbits of a connected reductive group over an algebraically closed field of\u0000characteristic 0. For classical groups we prove the characterization by a\u0000combinatorial computation, and for exceptional groups we check it with a\u0000computer. This characterization is needed in the theory of cuspidal sheaves on\u0000the stack of L-parameters in forthcoming work of the first two named authors.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187305","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}
{"title":"Categorifying Quiver Linking/Unlinking using CoHA Modules","authors":"Okke van Garderen","doi":"arxiv-2409.05605","DOIUrl":"https://doi.org/arxiv-2409.05605","url":null,"abstract":"The knots-quivers correspondence is a relation between knot invariants and\u0000enumerative invariants of quivers, which in particular translates the knot\u0000operations of linking and unlinking to a certain mutation operation on quivers.\u0000In this paper we show that the moduli spaces of a quiver and its\u0000linking/unlinking are naturally related, giving a purely representation\u0000theoretic interpretation of these operations. We obtain a relation between the\u0000cohomologies of these spaces which is moreover compatible with a natural action\u0000of the Cohomological Hall Algebra. The result is a categorification of quiver\u0000linking/unlinking at the level of CoHA modules.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187133","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}
{"title":"Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras","authors":"Aaron Chan, Osamu Iyama, Rene Marczinzik","doi":"arxiv-2409.05603","DOIUrl":"https://doi.org/arxiv-2409.05603","url":null,"abstract":"Auslander and Reiten called a finite dimensional algebra $A$ over a field\u0000Cohen-Macaulay if there is an $A$-bimodule $W$ which gives an equivalence\u0000between the category of finitely generated $A$-modules of finite projective\u0000dimension and the category of finitely generated $A$-modules of finite\u0000injective dimension. For example, Iwanaga-Gorenstein algebras and algebras with\u0000finitistic dimension zero on both sides are Cohen-Macaulay, and tensor products\u0000of Cohen-Macaulay algebras are again Cohen-Macaulay. They seem to be all of the\u0000known examples of Cohen-Macaulay algebras. In this paper, we give the first non-trivial class of Cohen-Macaulay algebras\u0000by showing that all contracted preprojective algebras of Dynkin type are\u0000Cohen-Macaulay. As a consequence, for each simple singularity $R$ and a maximal\u0000Cohen-Macaulay $R$-module $M$, the stable endomorphism algebra\u0000$underline{End}_R(M)$ is Cohen-Macaulay. We also give a negative answer to a\u0000question of Auslander-Reiten asking whether the category $CM A$ of\u0000Cohen-Macaulay $A$-modules coincides with the category of $d$-th syzygies,\u0000where $dge1$ is the injective dimension of $W$. In fact, if $A$ is a\u0000Cohen-Macaulay algebra that is additionally $d$-Gorenstein in the sense of\u0000Auslander, then $CM A$ always coincides with the category of $d$-th syzygies.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187121","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}
{"title":"Semiclassical functional calculus on nilpotent Lie groups and their compact nilmanifolds","authors":"Véronique Fischer, Søren Mikkelsen","doi":"arxiv-2409.05520","DOIUrl":"https://doi.org/arxiv-2409.05520","url":null,"abstract":"In this paper, we show that the semiclassical calculus recently developed on\u0000nilpotent Lie groups and nilmanifolds include the functional calculus of\u0000suitable subelliptic operators. Moreover, we obtain Weyl laws for these\u0000operators. Amongst these operators are sub-Laplacians in horizontal divergence\u0000form perturbed with a potential and their generalisations.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187129","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}
{"title":"Resolutions for Locally Analytic Representations","authors":"Shishir Agrawal, Matthias Strauch","doi":"arxiv-2409.05079","DOIUrl":"https://doi.org/arxiv-2409.05079","url":null,"abstract":"The purpose of this paper is to study resolutions of locally analytic\u0000representations of a $p$-adic reductive group $G$. Given a locally analytic\u0000representation $V$ of $G$, we modify the Schneider-Stuhler complex (originally\u0000defined for smooth representations) so as to give an `analytic' variant\u0000${mathcal S}^A_bullet(V)$. The representations in this complex are built out\u0000of spaces of analytic vectors $A_sigma(V)$ for compact open subgroups\u0000$U_sigma$, indexed by facets $sigma$ of the Bruhat-Tits building of $G$.\u0000These analytic representations (of compact open subgroups of $G$) are then\u0000resolved using the Chevalley-Eilenberg complex from the theory of Lie algebras.\u0000This gives rise to a resolution ${mathcal S}^{rm CE}_{q,bullet}(V)\u0000rightarrow {mathcal S}^A_q(V)$ for each representation ${mathcal S}^A_q(V)$\u0000in the analytic Schneider-Stuhler complex. In a last step we show that the\u0000family of representations ${mathcal S}^{rm CE}_{q,j}(V)$ can be given the\u0000structure of a Wall complex. The associated total complex ${mathcal S}^{rm\u0000CE}_bullet(V)$ has then the same homology as that of ${mathcal\u0000S}^A_bullet(V)$. If the latter is a resolution of $V$, then one can use\u0000${mathcal S}^{rm CE}_bullet(V)$ to find a complex which computes the\u0000extension group $underline{Ext}^n_G(V,W)$, provided $V$ and $W$ satisfy\u0000certain conditions which are satisfied when both are admissible locally\u0000analytic representations.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187126","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}
{"title":"Derived equivalences for the derived discrete algebras are standard","authors":"Grzegorz Bobinski, Tomasz Ciborski","doi":"arxiv-2409.05158","DOIUrl":"https://doi.org/arxiv-2409.05158","url":null,"abstract":"We prove that any derived equivalence between derived discrete algebras is\u0000standard, i.e. is isomorphic to the derived tensor product by a two-sided\u0000tilting complex.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187125","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}
{"title":"Counting points on character varieties","authors":"Masoud Kamgarpour, GyeongHyeon Nam, Bailey Whitbread, Stefano Giannini","doi":"arxiv-2409.04735","DOIUrl":"https://doi.org/arxiv-2409.04735","url":null,"abstract":"We count points on the character varieties associated with punctured surfaces\u0000and regular semisimple generic conjugacy classes in reductive groups. We find\u0000that the number of points are palindromic polynomials. This suggests a $P=W$\u0000conjecture for these varieties. We also count points on the corresponding\u0000additive character varieties and find that the number of points are also\u0000polynomials, which we conjecture have non-negative coefficients. These\u0000polynomials can be considered as the reductive analogues of the Kac polynomials\u0000of comet-shaped quivers.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187131","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}
{"title":"Monoidal categorification on some open Richardson varieties","authors":"Yingjin Bi","doi":"arxiv-2409.04715","DOIUrl":"https://doi.org/arxiv-2409.04715","url":null,"abstract":"In this paper, we investigate the cluster morphisms between cluster algebras\u0000and the functor of monoidal categories. Our findings demonstrate that there\u0000exists a monoidal categorification of the coordinate ring of the open\u0000Richardson variety $mathcal{R}_{w,v}$, where $w=vu$ and\u0000$ell(w)=ell(u)+ell(v)$. This result confirms a conjecture proposed by\u0000Kashiwara, Kim, Oh, and Park.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187127","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}
David Ben-Zvi, Yiannis Sakellaridis, Akshay Venkatesh
{"title":"Relative Langlands Duality","authors":"David Ben-Zvi, Yiannis Sakellaridis, Akshay Venkatesh","doi":"arxiv-2409.04677","DOIUrl":"https://doi.org/arxiv-2409.04677","url":null,"abstract":"We propose a duality in the relative Langlands program. This duality pairs a\u0000Hamiltonian space for a group $G$ with a Hamiltonian space under its dual group\u0000$check{G}$, and recovers at a numerical level the relationship between a\u0000period on $G$ and an $L$-function attached to $check{G}$; it is an arithmetic\u0000analog of the electric-magnetic duality of boundary conditions in\u0000four-dimensional supersymmetric Yang-Mills theory.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187128","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
