Pub Date : 2018-10-05DOI: 10.1142/s0219498822500128
Eneilson Campos, G. Martini, G. Fonseca
In this work the notions of partial action of a weak Hopf algebra on a coalgebra and partial action of a groupoid on a coalgebra will be introduced, just as some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of partial action on a coalgebra and partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.
{"title":"Partial actions of weak Hopf algebras on coalgebras","authors":"Eneilson Campos, G. Martini, G. Fonseca","doi":"10.1142/s0219498822500128","DOIUrl":"https://doi.org/10.1142/s0219498822500128","url":null,"abstract":"In this work the notions of partial action of a weak Hopf algebra on a coalgebra and partial action of a groupoid on a coalgebra will be introduced, just as some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of partial action on a coalgebra and partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129711776","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 $H$ be the Iwahori-Hecke algebra and let $J$ be Lusztig's asymptotic Hecke algebra, both specialized to type $tilde{A}_1$. For $mathrm{SL}_2$, when the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed recently that a completion of $H$ has codimension two as a subalgebra of a completion of $J$, and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis ${t_w}$ of $J$, and further invert the canonical isomorphism between the completions of $H$ and $J$, obtaining explicit formulas for the each basis element $t_w$ in terms of the basis $T_w$ of $H$. We conjecture some properties of this expansion for more general groups. We conclude by using our formulas to prove that $J$ acts on the Schwartz space of the basic affine space of $mathrm{SL}_2$, and produce some formulas for this action.
{"title":"On Lusztig’s asymptotic Hecke algebra for 𝑆𝐿₂","authors":"Stefan Dawydiak","doi":"10.1090/proc/15259","DOIUrl":"https://doi.org/10.1090/proc/15259","url":null,"abstract":"Let $H$ be the Iwahori-Hecke algebra and let $J$ be Lusztig's asymptotic Hecke algebra, both specialized to type $tilde{A}_1$. For $mathrm{SL}_2$, when the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed recently that a completion of $H$ has codimension two as a subalgebra of a completion of $J$, and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis ${t_w}$ of $J$, and further invert the canonical isomorphism between the completions of $H$ and $J$, obtaining explicit formulas for the each basis element $t_w$ in terms of the basis $T_w$ of $H$. We conjecture some properties of this expansion for more general groups. We conclude by using our formulas to prove that $J$ acts on the Schwartz space of the basic affine space of $mathrm{SL}_2$, and produce some formulas for this action.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132892559","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}
The character theory of finite groups has numerous basic questions that are often already quite involved: enumerating of irreducible characters, their character formulas, point-wise product decompositions, and restriction/induction between groups. A supercharacter theory is a framework for simplifying the character theory of a finite group, while ideally not losing all important information. This paper studies one such theory that straddles the gap between retaining valuable group information while reducing the above fundamental questions to more combinatorial lattice constructions.
{"title":"The structure of normal lattice supercharacter theories","authors":"F. Aliniaeifard, N. Thiem","doi":"10.5802/alco.126","DOIUrl":"https://doi.org/10.5802/alco.126","url":null,"abstract":"The character theory of finite groups has numerous basic questions that are often already quite involved: enumerating of irreducible characters, their character formulas, point-wise product decompositions, and restriction/induction between groups. A supercharacter theory is a framework for simplifying the character theory of a finite group, while ideally not losing all important information. This paper studies one such theory that straddles the gap between retaining valuable group information while reducing the above fundamental questions to more combinatorial lattice constructions.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"3 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121081159","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 : 2018-09-27DOI: 10.1007/978-3-030-82007-7_5
A. Braverman, M. Finkelberg
{"title":"A quasi-coherent description of the the category of D-mod(Gr$_{GL(n)}$)","authors":"A. Braverman, M. Finkelberg","doi":"10.1007/978-3-030-82007-7_5","DOIUrl":"https://doi.org/10.1007/978-3-030-82007-7_5","url":null,"abstract":"","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123725778","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}
The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis indexed by set-partition tableaux such that diagrams in the algebra act combinatorially on tableaux. The first representation is analogous to the Gelfand model and the second is a generalization of Young's natural representation of the symmetric group on standard tableaux. The methods of this paper work uniformly for the partition algebra and its diagram subalgebras. As an application, we express the characters of each of these algebras as nonnegative integer combinations of symmetric group characters whose coefficients count fixed points under conjugation.
{"title":"Set-partition tableaux and representations of diagram algebras","authors":"Tom Halverson, T. Jacobson","doi":"10.5802/alco.102","DOIUrl":"https://doi.org/10.5802/alco.102","url":null,"abstract":"The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis indexed by set-partition tableaux such that diagrams in the algebra act combinatorially on tableaux. The first representation is analogous to the Gelfand model and the second is a generalization of Young's natural representation of the symmetric group on standard tableaux. The methods of this paper work uniformly for the partition algebra and its diagram subalgebras. As an application, we express the characters of each of these algebras as nonnegative integer combinations of symmetric group characters whose coefficients count fixed points under conjugation.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130834506","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 : 2018-08-20DOI: 10.17323/1609-4514-2021-21-2-427-442
P. Pyatov, A. Trofimova
We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.
{"title":"Representations of Finite-Dimensional Quotient Algebras of the 3-String Braid Group","authors":"P. Pyatov, A. Trofimova","doi":"10.17323/1609-4514-2021-21-2-427-442","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-2-427-442","url":null,"abstract":"We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114318566","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 study collections of additive categories $mathcal{M}(G)$, indexed by finite groups $G$ and related by induction and restriction in a way that categorifies usual Mackey functors. We call them `Mackey 2-functors'. We provide a large collection of examples in particular thanks to additive derivators. We prove the first properties of Mackey 2-functors, including separable monadicity of restriction to subgroups. We then isolate the initial such structure, leading to what we call `Mackey 2-motives'. We also exhibit a convenient calculus of morphisms in Mackey 2-motives, by means of string diagrams. Finally, we show that the 2-endomorphism ring of the identity of $G$ in this 2-category of Mackey 2-motives is isomorphic to the so-called crossed Burnside ring of $G$.
{"title":"Mackey 2-Functors and Mackey 2-Motives","authors":"Paul Balmer, Ivo Dell’Ambrogio","doi":"10.4171/209","DOIUrl":"https://doi.org/10.4171/209","url":null,"abstract":"We study collections of additive categories $mathcal{M}(G)$, indexed by finite groups $G$ and related by induction and restriction in a way that categorifies usual Mackey functors. We call them `Mackey 2-functors'. We provide a large collection of examples in particular thanks to additive derivators. We prove the first properties of Mackey 2-functors, including separable monadicity of restriction to subgroups. We then isolate the initial such structure, leading to what we call `Mackey 2-motives'. We also exhibit a convenient calculus of morphisms in Mackey 2-motives, by means of string diagrams. Finally, we show that the 2-endomorphism ring of the identity of $G$ in this 2-category of Mackey 2-motives is isomorphic to the so-called crossed Burnside ring of $G$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133467005","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 describe all two sided ideals of a cyclotomic rational Cherednik algebra $H_mathbf{c}$ and its spherical subalgebra $eH_mathbf{c} e$ with a Weil generic aspherical parameter $mathbf{c}$, and further describe the simple modules in the category $mathcal{O}^{sph}_mathbf{c}$ . The main tools we use are categorical Kac-Moody actions on catogories $mathcal{O}_mathbf{c}$ and restriction functors for Harish-Chandra bimodules.
{"title":"Representations of cyclotomic rational Cherednik algebras with aspherical parameters","authors":"Huijun Zhao","doi":"10.17760/d20291522","DOIUrl":"https://doi.org/10.17760/d20291522","url":null,"abstract":"In this article, we describe all two sided ideals of a cyclotomic rational Cherednik algebra $H_mathbf{c}$ and its spherical subalgebra $eH_mathbf{c} e$ with a Weil generic aspherical parameter $mathbf{c}$, and further describe the simple modules in the category $mathcal{O}^{sph}_mathbf{c}$ . The main tools we use are categorical Kac-Moody actions on catogories $mathcal{O}_mathbf{c}$ and restriction functors for Harish-Chandra bimodules.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"193 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121101046","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}
Admissible W-graphs were defined and combinatorially characterised by Stembridge in reference [12]. The theory of admissible W-graphs was motivated by the need to construct W-graphs for Kazhdan-Lusztig cells, which play an important role in the representation theory of Hecke algebras, without computing Kazhdan-Lusztig polynomials. In this paper, we shall show that type A-admissible W-cells are Kazhdan-Lusztig as conjectured by Stembridge in his original paper.
{"title":"Type $A$ admissible cells are Kazhdan–Lusztig","authors":"V. M. Nguyen","doi":"10.5802/alco.91","DOIUrl":"https://doi.org/10.5802/alco.91","url":null,"abstract":"Admissible W-graphs were defined and combinatorially characterised by Stembridge in reference [12]. The theory of admissible W-graphs was motivated by the need to construct W-graphs for Kazhdan-Lusztig cells, which play an important role in the representation theory of Hecke algebras, without computing Kazhdan-Lusztig polynomials. In this paper, we shall show that type A-admissible W-cells are Kazhdan-Lusztig as conjectured by Stembridge in his original paper.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130548547","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 the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group.
{"title":"Higher deformations of Lie algebra representations I","authors":"Matthew Westaway","doi":"10.2969/jmsj/81188118","DOIUrl":"https://doi.org/10.2969/jmsj/81188118","url":null,"abstract":"In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133536552","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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