The convolution ring $K^{GL_n(mathcal{O})rtimesmathbb{C}^times}(mathrm{Gr}_{GL_n})$ was identified with a quantum unipotent cell of the loop group $LSL_2$ in [Cautis-Williams, arXiv:1801.08111]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.
{"title":"Coherent IC-sheaves on type 𝐴_{𝑛} affine Grassmannians and dual canonical basis of affine type 𝐴₁","authors":"M. Finkelberg, Ryo Fujita","doi":"10.1090/ERT/558","DOIUrl":"https://doi.org/10.1090/ERT/558","url":null,"abstract":"The convolution ring $K^{GL_n(mathcal{O})rtimesmathbb{C}^times}(mathrm{Gr}_{GL_n})$ was identified with a quantum unipotent cell of the loop group $LSL_2$ in [Cautis-Williams, arXiv:1801.08111]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131608330","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}
Pub Date : 2019-01-11DOI: 10.22108/IJGT.2019.112277.1490
Donnie Munyao Kasyoki, P. Oleche
Let $G$ be a finite group and $text{cd}(G)$ denote the character degree set for $G$. The prime graph $Delta(G)$ is a simple graph whose vertex set consists of prime divisors of elements in $text{cd}(G)$, denoted $rho(G)$. Two primes $p,qin rho(G)$ are adjacent in $Delta(G)$ if and only if $pq|a$ for some $ain text{cd}(G)$. We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group.
{"title":"4-Regular prime graphs of nonsolvable groups","authors":"Donnie Munyao Kasyoki, P. Oleche","doi":"10.22108/IJGT.2019.112277.1490","DOIUrl":"https://doi.org/10.22108/IJGT.2019.112277.1490","url":null,"abstract":"Let $G$ be a finite group and $text{cd}(G)$ denote the character degree set for $G$. The prime graph $Delta(G)$ is a simple graph whose vertex set consists of prime divisors of elements in $text{cd}(G)$, denoted $rho(G)$. Two primes $p,qin rho(G)$ are adjacent in $Delta(G)$ if and only if $pq|a$ for some $ain text{cd}(G)$. We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132319038","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}
Pub Date : 2019-01-07DOI: 10.2140/PJM.2020.306.587
Dongwen Liu, Zhicheng Wang
S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations.
{"title":"Remarks on the theta correspondence over finite fields","authors":"Dongwen Liu, Zhicheng Wang","doi":"10.2140/PJM.2020.306.587","DOIUrl":"https://doi.org/10.2140/PJM.2020.306.587","url":null,"abstract":"S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114293171","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 show that the motive of a Springer fiber is pure Tate. We then consider a category of equivariant Springer motives on the nilpotent cone and construct an equivalence to the derived category of graded modules over the graded affine Hecke algebra.
{"title":"Springer motives","authors":"J. Eberhardt","doi":"10.1090/proc/15290","DOIUrl":"https://doi.org/10.1090/proc/15290","url":null,"abstract":"We show that the motive of a Springer fiber is pure Tate. We then consider a category of equivariant Springer motives on the nilpotent cone and construct an equivalence to the derived category of graded modules over the graded affine Hecke algebra.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127409564","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 defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(mathbb{C}^n)^{otimes k}$, where $mathbb{C}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p =1$, the corresponding partition algebra was studied by Orellana. We also establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $(mathbb{C}^n)^{otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras. We conclude the paper by giving Jucys-Murphy elements of Tanabe algebras and their actions on the Gelfand-Tsetlin basis, determined by this multiplicity free tower, of irreducible modules.
{"title":"On representation theory of partition algebras for complex reflection groups","authors":"Ashish Mishra, S. Srivastava","doi":"10.5802/ALCO.97","DOIUrl":"https://doi.org/10.5802/ALCO.97","url":null,"abstract":"This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(mathbb{C}^n)^{otimes k}$, where $mathbb{C}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p =1$, the corresponding partition algebra was studied by Orellana. We also establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $(mathbb{C}^n)^{otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras. We conclude the paper by giving Jucys-Murphy elements of Tanabe algebras and their actions on the Gelfand-Tsetlin basis, determined by this multiplicity free tower, of irreducible modules.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131418792","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}
Pub Date : 2018-12-10DOI: 10.1142/S179304212150024X
J. Smith
Let $F$ be a nonarchimedean local field of characteristic zero and odd residual characteristic. Let $X$ be the $p$-adic symmetric space $X = H backslash G$, where $G = mathbf{GL}_{2n}(F)$ and $H = mathbf{GL}_n(F) times mathbf{GL}_n(F)$. We verify a conjecture of Sakellaridis and Venkatesh on the Langlands parameters of certain representations in the discrete spectrum of $X$.
设$F$为特征为零且残差特征为奇的非阿基米德局部域。设$X$为$p$进进对称空间$X = H 反斜杠G$，其中$G = mathbf{GL}_{2n}(F)$， $H = mathbf{GL}_n(F) 乘以mathbf{GL}_n(F)$。我们验证了Sakellaridis和Venkatesh关于X离散谱中某些表示的朗兰兹参数的猜想。
{"title":"Linear periods and distinguished local parameters","authors":"J. Smith","doi":"10.1142/S179304212150024X","DOIUrl":"https://doi.org/10.1142/S179304212150024X","url":null,"abstract":"Let $F$ be a nonarchimedean local field of characteristic zero and odd residual characteristic. Let $X$ be the $p$-adic symmetric space $X = H backslash G$, where $G = mathbf{GL}_{2n}(F)$ and $H = mathbf{GL}_n(F) times mathbf{GL}_n(F)$. We verify a conjecture of Sakellaridis and Venkatesh on the Langlands parameters of certain representations in the discrete spectrum of $X$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132796014","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 compute special unipotent Arthur packets for real reductive groups in many cases. We list the cases that lead to incomplete answers, and in those cases, provide a suitable set of representations that could lead to a complete description of the special Arthur packet. In the process of achieving this goal we classify theta forms of a given even complex nilpotent orbit, and find methods to compute the associated varieties of irreducible group representations.
{"title":"Special Unipotent Arthur Packets for Real Reductive Groups","authors":"J. Fernandes","doi":"10.13016/QMF9-NTWP","DOIUrl":"https://doi.org/10.13016/QMF9-NTWP","url":null,"abstract":"We compute special unipotent Arthur packets for real reductive groups in many cases. We list the cases that lead to incomplete answers, and in those cases, provide a suitable set of representations that could lead to a complete description of the special Arthur packet. In the process of achieving this goal we classify theta forms of a given even complex nilpotent orbit, and find methods to compute the associated varieties of irreducible group representations.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"199 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116185195","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}
Pub Date : 2018-11-01DOI: 10.2140/PJM.2020.306.557
Rosanna Laking, Jorge Vit'oria
We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy colimits and further conditions involving the notion of purity. In particular, we provide sufficient closure conditions for a subcategory of a compactly generated algebraic triangulated category to be a torsion class. Finally we explore applications of the previous results to the theory of recollements.
{"title":"Definability and approximations in triangulated categories","authors":"Rosanna Laking, Jorge Vit'oria","doi":"10.2140/PJM.2020.306.557","DOIUrl":"https://doi.org/10.2140/PJM.2020.306.557","url":null,"abstract":"We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy colimits and further conditions involving the notion of purity. In particular, we provide sufficient closure conditions for a subcategory of a compactly generated algebraic triangulated category to be a torsion class. Finally we explore applications of the previous results to the theory of recollements.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121232095","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":"Deformed mesh algebras of Dynkin type $mathbb{F}_4$","authors":"J. Białkowski","doi":"10.4064/CM126-2-6","DOIUrl":"https://doi.org/10.4064/CM126-2-6","url":null,"abstract":"We prove that every deformed mesh algebra of type $mathbb{F}_4$ is isomorphic to the canonical mesh algebra of type $mathbb{F}_4$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132884194","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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