Mahir Bilen Can, S. Senthamarai Kannan, Pinakinath Saha
Horospherical Schubert varieties are determined. It is shown that the stabilizer of an arbitrary point in a Schubert variety is a strongly solvable algebraic group. The connectedness of this stabilizer subgroup is discussed. Moreover, a new family of spherical varieties, called doubly spherical varieties, is introduced. It is shown that every nearly toric Schubert variety is doubly spherical.
{"title":"From Schubert Varieties to Doubly-Spherical Varieties","authors":"Mahir Bilen Can, S. Senthamarai Kannan, Pinakinath Saha","doi":"arxiv-2409.04879","DOIUrl":"https://doi.org/arxiv-2409.04879","url":null,"abstract":"Horospherical Schubert varieties are determined. It is shown that the\u0000stabilizer of an arbitrary point in a Schubert variety is a strongly solvable\u0000algebraic group. The connectedness of this stabilizer subgroup is discussed.\u0000Moreover, a new family of spherical varieties, called doubly spherical\u0000varieties, is introduced. It is shown that every nearly toric Schubert variety\u0000is doubly spherical.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187130","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}
Geiss, Leclerc and Schr"oer introduced a class of 1-Iwanaga-Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. They also proved that indecomposable rigid $H$-modules of finite projective dimension are in bijection with non-initial cluster variables of the corresponding Fomin-Zelevinsky cluster algebra. In this article, we prove in all affine types that their conjectural Caldero-Chapoton type formula on these modules coincide with the Laurent expression of cluster variables. By taking generic Caldero-Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials.
{"title":"Generic bases of skew-symmetrizable affine type cluster algebras","authors":"Lang Mou, Xiuping Su","doi":"arxiv-2409.03954","DOIUrl":"https://doi.org/arxiv-2409.03954","url":null,"abstract":"Geiss, Leclerc and Schr\"oer introduced a class of 1-Iwanaga-Gorenstein\u0000algebras $H$ associated to symmetrizable Cartan matrices with acyclic\u0000orientations, generalizing the path algebras of acyclic quivers. They also\u0000proved that indecomposable rigid $H$-modules of finite projective dimension are\u0000in bijection with non-initial cluster variables of the corresponding\u0000Fomin-Zelevinsky cluster algebra. In this article, we prove in all affine types\u0000that their conjectural Caldero-Chapoton type formula on these modules coincide\u0000with the Laurent expression of cluster variables. By taking generic\u0000Caldero-Chapoton functions on varieties of modules of finite projective\u0000dimension, we obtain bases for affine type cluster algebras with full-rank\u0000coefficients containing all cluster monomials.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187136","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}
Wille Liu, Cheng-Chiang Tsai, Kari Vilonen, Ting Xue
We show in this paper that in the context of graded Lie algebras, all cuspidal character sheaves arise from a nearby-cycle construction followed by a Fourier--Sato transform in a very specific manner. Combined with results of the last two named authors, this completes the explicit description of cuspidal character sheaves for Vinberg's type I graded classical Lie algebras.
我们在本文中证明，在有级李代数的背景下，所有uspidal character sheaves都是以一种非常特殊的方式由邻近循环构造和傅里叶--萨托变换（Fourier--Sato transform）产生的。结合前两位作者的研究成果，本文完成了对文伯格 I 型梯度经典列阵的无顶角特征卷的明确描述。
{"title":"Cuspidal character sheaves on graded Lie algebras","authors":"Wille Liu, Cheng-Chiang Tsai, Kari Vilonen, Ting Xue","doi":"arxiv-2409.04030","DOIUrl":"https://doi.org/arxiv-2409.04030","url":null,"abstract":"We show in this paper that in the context of graded Lie algebras, all\u0000cuspidal character sheaves arise from a nearby-cycle construction followed by a\u0000Fourier--Sato transform in a very specific manner. Combined with results of the\u0000last two named authors, this completes the explicit description of cuspidal\u0000character sheaves for Vinberg's type I graded classical Lie algebras.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187134","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 Lie group. Consider a nilpotent element $ein mathfrak{g}$. Let $Z_G(e)$ be the centralizer of $e$ in $G$, and let $A_e:= Z_G(e)/Z_G(e)^{o}$ be its component group. Write $text{Irr}(mathcal{B}_e)$ for the set of irreducible components of the Springer fiber $mathcal{B}_e$. We have an action of $A_e$ on $text{Irr}(mathcal{B}_e)$. When $mathfrak{g}$ is exceptional, we give an explicit description of $text{Irr}(mathcal{B}_e)$ as an $A_e$-set. For $mathfrak{g}$ of classical type, we describe the stabilizers for the $A_e$-action. With this description, we prove a conjecture of Lusztig and Sommers.
{"title":"The action of component groups on irreducible components of Springer fibers","authors":"Do Kien Hoang","doi":"arxiv-2409.04076","DOIUrl":"https://doi.org/arxiv-2409.04076","url":null,"abstract":"Let $G$ be a simple Lie group. Consider a nilpotent element $ein\u0000mathfrak{g}$. Let $Z_G(e)$ be the centralizer of $e$ in $G$, and let $A_e:=\u0000Z_G(e)/Z_G(e)^{o}$ be its component group. Write $text{Irr}(mathcal{B}_e)$\u0000for the set of irreducible components of the Springer fiber $mathcal{B}_e$. We\u0000have an action of $A_e$ on $text{Irr}(mathcal{B}_e)$. When $mathfrak{g}$ is\u0000exceptional, we give an explicit description of $text{Irr}(mathcal{B}_e)$ as\u0000an $A_e$-set. For $mathfrak{g}$ of classical type, we describe the stabilizers\u0000for the $A_e$-action. With this description, we prove a conjecture of Lusztig\u0000and Sommers.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187132","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 establish a derived geometric Satake equivalence for the real group $G_{mathbb R}=PSO(2n-1,1)$ (resp. $PE_6(F_4)$), to be called the Lorentzian Satake equivalence (resp. Octonionic Satake equivalence). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the splitting rank symmetric variety $X=PSO_{2n}/SO_{2n-1}$ (resp. $PE_6/F_4$). As an application, we compute the stalks of the $text{IC}$-complexes for spherical orbit closures in the real affine Grassmannian for $G_{mathbb R}$ and the loop space of $X$. We show the stalks are given by the Kostka-Foulkes polynomials for $GL_2$ (resp. $GL_3$) but with $q$ replaced by $q^{n-1}$ (resp. $q^4$).
{"title":"Lorentzian and Octonionic Satake equivalence","authors":"Tsao-Hsien Chen, John O'Brien","doi":"arxiv-2409.03969","DOIUrl":"https://doi.org/arxiv-2409.03969","url":null,"abstract":"We establish a derived geometric Satake equivalence for the real group\u0000$G_{mathbb R}=PSO(2n-1,1)$ (resp. $PE_6(F_4)$), to be called the Lorentzian\u0000Satake equivalence (resp. Octonionic Satake equivalence). By applying the\u0000real-symmetric correspondence for affine Grassmannians, we obtain a derived\u0000geometric Satake equivalence for the splitting rank symmetric variety\u0000$X=PSO_{2n}/SO_{2n-1}$ (resp. $PE_6/F_4$). As an application, we compute the\u0000stalks of the $text{IC}$-complexes for spherical orbit closures in the real\u0000affine Grassmannian for $G_{mathbb R}$ and the loop space of $X$. We show the\u0000stalks are given by the Kostka-Foulkes polynomials for $GL_2$ (resp. $GL_3$)\u0000but with $q$ replaced by $q^{n-1}$ (resp. $q^4$).","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187135","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}
G. Navarro and P. H. Tiep, among others, have studied groups with few rational conjugacy classes or few rational irreducible characters. In this paper we look at the opposite extreme. Let $G$ be a finite group. Given a conjugacy class $K$ of $G$, we say it is irrational if there is some $chi in operatorname{Irr}(G)$ such that $chi(K) not in mathbb{Q}$. One of our main results shows that, when $G$ contains at most $5$ irrational conjugacy classes, then $|operatorname{Irr}_{mathbb{Q}} (G)| = | operatorname{cl}_{mathbb{Q}} (G)|$. This suggests some duality with the known results and open questions on groups with few rational irreducible characters.
G. Navarro 和 P. H. Tiep 等人研究了具有少数有理共轭类或少数有理不可还原符的群。在本文中，我们将研究相反的极端。假设 $G$ 是一个有限群。给定 $G$ 的共轭类 $K$，如果存在某个 $chi inoperatorname{Irr}(G)$ 使得 $chi(K) not in mathbb{Q}$ ，我们就说它是无理的。我们的一个主要结果表明，当 $G$ 包含最多 5$ 个无理共轭类时，$|operatorname{Irr}_{mathbb{Q}}.(G)| = | operatorname{cl}_{mathbb{Q}}(G)|$.这表明，在具有少量有理不可还原字符的群上，已知的结果和悬而未决的问题具有一定的对偶性。
{"title":"On groups with at most five irrational conjugacy classes","authors":"Gabriel de Arêa Leão Souza","doi":"arxiv-2409.03539","DOIUrl":"https://doi.org/arxiv-2409.03539","url":null,"abstract":"G. Navarro and P. H. Tiep, among others, have studied groups with few\u0000rational conjugacy classes or few rational irreducible characters. In this\u0000paper we look at the opposite extreme. Let $G$ be a finite group. Given a\u0000conjugacy class $K$ of $G$, we say it is irrational if there is some $chi in\u0000operatorname{Irr}(G)$ such that $chi(K) not in mathbb{Q}$. One of our main\u0000results shows that, when $G$ contains at most $5$ irrational conjugacy classes,\u0000then $|operatorname{Irr}_{mathbb{Q}} (G)| = | operatorname{cl}_{mathbb{Q}}\u0000(G)|$. This suggests some duality with the known results and open questions on\u0000groups with few rational irreducible characters.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187142","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 prove almost sure strong asymptotic freeness of i.i.d. random unitaries with the following law: sample a Haar unitary matrix of dimension $n$ and then send this unitary into an irreducible representation of $U(n)$. The strong convergence holds as long as the irreducible representation arises from a pair of partitions of total size at most $n^{frac{1}{24}-varepsilon}$ and is uniform in this regime. Previously this was known for partitions of total size up to $asymplog n/loglog n$ by a result of Bordenave and Collins.
{"title":"Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations","authors":"Michael Magee, Mikael de la Salle","doi":"arxiv-2409.03626","DOIUrl":"https://doi.org/arxiv-2409.03626","url":null,"abstract":"We prove almost sure strong asymptotic freeness of i.i.d. random unitaries\u0000with the following law: sample a Haar unitary matrix of dimension $n$ and then\u0000send this unitary into an irreducible representation of $U(n)$. The strong\u0000convergence holds as long as the irreducible representation arises from a pair\u0000of partitions of total size at most $n^{frac{1}{24}-varepsilon}$ and is\u0000uniform in this regime. Previously this was known for partitions of total size up to $asymplog\u0000n/loglog n$ by a result of Bordenave and Collins.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187139","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 present a novel axiomatic framework for establishing horizontal norm relations in Euler systems that are built from pushforwards of classes in the motivic cohomology of Shimura varieties. This framework is uniformly applicable to the Euler systems of both algebraic cycles and Eisenstein classes. It also applies to non-spherical pairs of groups that fail to satisfy a local multiplicity one hypothesis, and thus lie beyond the reach of existing methods. A key application of this work is the construction of an Euler system for the spinor Galois representations arising in the cohomology of Siegel modular varieties of genus three, which is undertaken in two companion articles.
{"title":"On constructing zeta elements for Shimura varieties","authors":"Syed Waqar Ali Shah","doi":"arxiv-2409.03517","DOIUrl":"https://doi.org/arxiv-2409.03517","url":null,"abstract":"We present a novel axiomatic framework for establishing horizontal norm\u0000relations in Euler systems that are built from pushforwards of classes in the\u0000motivic cohomology of Shimura varieties. This framework is uniformly applicable\u0000to the Euler systems of both algebraic cycles and Eisenstein classes. It also\u0000applies to non-spherical pairs of groups that fail to satisfy a local\u0000multiplicity one hypothesis, and thus lie beyond the reach of existing methods.\u0000A key application of this work is the construction of an Euler system for the\u0000spinor Galois representations arising in the cohomology of Siegel modular\u0000varieties of genus three, which is undertaken in two companion articles.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187143","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 establish abstract horizontal norm relations involving the unramified Hecke-Frobenius polynomials that correspond under the Satake isomorhpism to the degree eight spinor $L$-factors of $ mathrm{GSp}_{6} $. These relations apply to classes in the degree seven motivic cohomology of the Siegel modular sixfold obtained via Gysin pushforwards of Beilinson's Eisenstein symbol pulled back on one copy in a triple product of modular curves. The proof is based on a novel approach that circumvents the failure of the so-called multiplicity one hypothesis in our setting, which precludes the applicability of an existing technique. In a sequel, we combine our result with the previously established vertical norm relations for these classes to obtain new Euler systems for the eight dimensional Galois representations associated with certain non-endoscopic cohomological cuspidal automorphic representations of $ mathrm{GSp}_{6} $.
{"title":"Horizontal norm compatibility of cohomology classes for $mathrm{GSp}_{6}$","authors":"Syed Waqar Ali Shah","doi":"arxiv-2409.03738","DOIUrl":"https://doi.org/arxiv-2409.03738","url":null,"abstract":"We establish abstract horizontal norm relations involving the unramified\u0000Hecke-Frobenius polynomials that correspond under the Satake isomorhpism to the\u0000degree eight spinor $L$-factors of $ mathrm{GSp}_{6} $. These relations apply\u0000to classes in the degree seven motivic cohomology of the Siegel modular sixfold\u0000obtained via Gysin pushforwards of Beilinson's Eisenstein symbol pulled back on\u0000one copy in a triple product of modular curves. The proof is based on a novel\u0000approach that circumvents the failure of the so-called multiplicity one\u0000hypothesis in our setting, which precludes the applicability of an existing\u0000technique. In a sequel, we combine our result with the previously established\u0000vertical norm relations for these classes to obtain new Euler systems for the\u0000eight dimensional Galois representations associated with certain non-endoscopic\u0000cohomological cuspidal automorphic representations of $ mathrm{GSp}_{6} $.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187140","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 first prove that the maximal ideal of the universal affine vertex operator algebra $V^k(sl_n)$ for $k=-n+frac{n-1}{q}$ is generated by two singular vectors of conformal weight $3q$ if $n=3$, and by one singular vector of conformal weight $2q$ if $ngeq 4$. We next determine the associated varieties of the simple vertex operator algebras $L_k(sl_3)$ for all the non-admissible levels $k=-3+frac{2}{2m+1}$, $mgeq 0$. The varieties of the associated simple affine $W$-algebras $W_k(sl_3,f)$, for nilpotent elements $f$ of $sl_3$, are also determined.
{"title":"Associated varieties of simple affine VOAs $L_k(sl_3)$ and $W$-algebras $W_k(sl_3,f)$","authors":"Cuipo Jiang, Jingtian Song","doi":"arxiv-2409.03552","DOIUrl":"https://doi.org/arxiv-2409.03552","url":null,"abstract":"In this paper we first prove that the maximal ideal of the universal affine\u0000vertex operator algebra $V^k(sl_n)$ for $k=-n+frac{n-1}{q}$ is generated by\u0000two singular vectors of conformal weight $3q$ if $n=3$, and by one singular\u0000vector of conformal weight $2q$ if $ngeq 4$. We next determine the associated\u0000varieties of the simple vertex operator algebras $L_k(sl_3)$ for all the\u0000non-admissible levels $k=-3+frac{2}{2m+1}$, $mgeq 0$. The varieties of the\u0000associated simple affine $W$-algebras $W_k(sl_3,f)$, for nilpotent elements $f$\u0000of $sl_3$, are also determined.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187343","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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