Pub Date : 2020-08-18DOI: 10.2140/pjm.2021.311.455
Alistair Savage, John C. Stuart
To any Frobenius superalgebra $A$ we associate towers of Frobenius nilCoxeter algebras and Frobenius nilHecke algebras. These act naturally, via Frobeinus divided difference operators, on Frobenius polynomial algebras. When $A$ is the ground ring, our algebras recover the classical nilCoxeter and nilHecke algebras. When $A$ is the two-dimensional Clifford algebra, they are Morita equivalent to the odd nilCoxeter and odd nilHecke algebras.
{"title":"Frobenius nil-Hecke algebras","authors":"Alistair Savage, John C. Stuart","doi":"10.2140/pjm.2021.311.455","DOIUrl":"https://doi.org/10.2140/pjm.2021.311.455","url":null,"abstract":"To any Frobenius superalgebra $A$ we associate towers of Frobenius nilCoxeter algebras and Frobenius nilHecke algebras. These act naturally, via Frobeinus divided difference operators, on Frobenius polynomial algebras. When $A$ is the ground ring, our algebras recover the classical nilCoxeter and nilHecke algebras. When $A$ is the two-dimensional Clifford algebra, they are Morita equivalent to the odd nilCoxeter and odd nilHecke algebras.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116601436","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 : 2020-08-16DOI: 10.2140/pjm.2021.311.369
R. Joshi, S. Spallone
We determine which orthogonal representations V of GL(n,q) lift to the double cover Pin(V ) of the orthogonal group O(V ). We cover all n and prime powers q, except for (n; q) =(3,4).
{"title":"Spinoriality of orthogonal representations of\u0000GLn(𝔽q)","authors":"R. Joshi, S. Spallone","doi":"10.2140/pjm.2021.311.369","DOIUrl":"https://doi.org/10.2140/pjm.2021.311.369","url":null,"abstract":"We determine which orthogonal representations V of GL(n,q) lift to the double cover Pin(V ) of the orthogonal group O(V ). We cover all n and prime powers q, except for (n; q) =(3,4).","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121918482","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 establish the inductive Alperin weight condition for the finite simple groups of Lie type $mathsf A$, contributing to the program to prove the Alperin weight conjecture by checking the inductive condition for all finite simple groups.
{"title":"Equivariant correspondences and the inductive Alperin weight condition for type $mathsf {A}$","authors":"Zhicheng Feng, Conghui Li, Jiping Zhang","doi":"10.1090/TRAN/8463","DOIUrl":"https://doi.org/10.1090/TRAN/8463","url":null,"abstract":"In this paper, we establish the inductive Alperin weight condition for the finite simple groups of Lie type $mathsf A$, contributing to the program to prove the Alperin weight conjecture by checking the inductive condition for all finite simple groups.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124377950","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 : 2020-08-10DOI: 10.1007/S00209-021-02746-2
S. Fishel, Stephen Griffeth, Elizabeth Manosalva
{"title":"Unitary representations of the Cherednik algebra: $V^*$-homology","authors":"S. Fishel, Stephen Griffeth, Elizabeth Manosalva","doi":"10.1007/S00209-021-02746-2","DOIUrl":"https://doi.org/10.1007/S00209-021-02746-2","url":null,"abstract":"","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129722621","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 the authors introduce an analog of the nilpotent cone, ${mathcal N}$, for a classical Lie superalgebra, ${mathfrak g}$, that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, ${mathfrak g}={mathfrak g}_{bar{0}}oplus {mathfrak g}_{bar{1}}$ with $text{Lie }G_{bar{0}}={mathfrak g}_{bar{0}}$, it is shown that there are finitely many $G_{bar{0}}$-orbits on ${mathcal N}$. Later the authors prove that the Duflo-Serganova commuting variety, ${mathcal X}$, is contained in ${mathcal N}$ for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.
{"title":"The nilpotent cone for classical Lie superalgebras","authors":"L. A. Jenkins, D. Nakano","doi":"10.1090/PROC/15599","DOIUrl":"https://doi.org/10.1090/PROC/15599","url":null,"abstract":"In this paper the authors introduce an analog of the nilpotent cone, ${mathcal N}$, for a classical Lie superalgebra, ${mathfrak g}$, that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, ${mathfrak g}={mathfrak g}_{bar{0}}oplus {mathfrak g}_{bar{1}}$ with $text{Lie }G_{bar{0}}={mathfrak g}_{bar{0}}$, it is shown that there are finitely many $G_{bar{0}}$-orbits on ${mathcal N}$. Later the authors prove that the Duflo-Serganova commuting variety, ${mathcal X}$, is contained in ${mathcal N}$ for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114219200","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 work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Plucker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.
{"title":"With Wronskian through the Looking Glass","authors":"V. Gorbounov, V. Schechtman","doi":"10.3842/sigma.2021.001","DOIUrl":"https://doi.org/10.3842/sigma.2021.001","url":null,"abstract":"In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Plucker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"147 suppl_2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114300723","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 : 2020-06-30DOI: 10.1007/978-981-15-7775-8_2
Toshiyuki Kobayashi, B. Speh
{"title":"A Hidden Symmetry of a Branching Law","authors":"Toshiyuki Kobayashi, B. Speh","doi":"10.1007/978-981-15-7775-8_2","DOIUrl":"https://doi.org/10.1007/978-981-15-7775-8_2","url":null,"abstract":"","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115024988","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}
A conjecture going back to the eighties claims that there are no non-trivial self-extensions of irreducible modules over symmetric groups if the characteristic of the ground field is not equal to $2$. We obtain some partial positive results on this conjecture.
{"title":"On self-extensions of irreducible modules over symmetric groups","authors":"Haralampos Geranios, A. Kleshchev, Lucia Morotti","doi":"10.1090/tran/8566","DOIUrl":"https://doi.org/10.1090/tran/8566","url":null,"abstract":"A conjecture going back to the eighties claims that there are no non-trivial self-extensions of irreducible modules over symmetric groups if the characteristic of the ground field is not equal to $2$. We obtain some partial positive results on this conjecture.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131781287","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}
Majorization is a partial order on real vectors which plays an important role in a variety of subjects, ranging from algebra and combinatorics to probability and statistics. In this paper, we consider a generalized notion of majorization associated to an arbitrary root system $Phi,$ and show that it admits a natural characterization in terms of the values of spherical functions on any Riemannian symmetric space with restricted root system $Phi.$
{"title":"Majorization and Spherical Functions","authors":"Colin S. McSwiggen, Jonathan Novak","doi":"10.1093/IMRN/RNAA390","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA390","url":null,"abstract":"Majorization is a partial order on real vectors which plays an important role in a variety of subjects, ranging from algebra and combinatorics to probability and statistics. In this paper, we consider a generalized notion of majorization associated to an arbitrary root system $Phi,$ and show that it admits a natural characterization in terms of the values of spherical functions on any Riemannian symmetric space with restricted root system $Phi.$","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126571498","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 studies scattered representations of $G = SO(2n+1, mathbb{C})$, $Sp(2n, mathbb{C})$ and $SO(2n, mathbb{C})$, which lies in the `core' of the unitary spectrum $G$ with nonzero Dirac cohomology. We describe the Zhelobenko parameters of these representations, count their cardinality, and determine their spin-lowest $K$-types. We also disprove a conjecture raised in 2015 asserting that the unitary dual can be obtained via parabolic induction from irreducible unitary representations with non-zero Dirac cohomology.
{"title":"Scattered Representations of Complex Classical Lie Groups","authors":"Chaoping Dong, K. Wong","doi":"10.1093/IMRN/RNAA388","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA388","url":null,"abstract":"This paper studies scattered representations of $G = SO(2n+1, mathbb{C})$, $Sp(2n, mathbb{C})$ and $SO(2n, mathbb{C})$, which lies in the `core' of the unitary spectrum $G$ with nonzero Dirac cohomology. We describe the Zhelobenko parameters of these representations, count their cardinality, and determine their spin-lowest $K$-types. We also disprove a conjecture raised in 2015 asserting that the unitary dual can be obtained via parabolic induction from irreducible unitary representations with non-zero Dirac cohomology.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115433140","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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