The generalized Gelfand–Graev characters of GLn(Fq)

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6406 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Scott Andrews ◽

Nathaniel Thiem

Keyword(s):

Finite Groups ◽

Type A ◽

General Linear ◽

Linear Groups ◽

General Linear Groups ◽

Representations Of Finite Groups ◽

International Audience ◽

Unipotent Representations ◽

Groups Of Lie Type

International audience Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, gener- alized Gelfand–Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's def- inition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand–Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand–Graev characters in terms of unipotent representations, thereby recovering the Kostka–Foulkes polynomials as multiplicities.

Download Full-text

James Alexander Green. 26 February 1926—7 April 2014

Biographical Memoirs of Fellows of the Royal Society ◽

10.1098/rsbm.2019.0012 ◽

2019 ◽

Vol 67 ◽

pp. 173-190

Author(s):

Stephen Donkin ◽

Karin Erdmann

Keyword(s):

Finite Groups ◽

Royal Society ◽

Great Influence ◽

General Linear ◽

Fundamental Result ◽

Linear Groups ◽

Theoretic Approach ◽

General Linear Groups ◽

Polynomial Representations ◽

Representations Of Finite Groups

James Alexander Green, known as Sandy, was a mathematician of great influence and distinction. He was an algebraist, famous for his work on modular representations of finite groups, and the development of the theory of polynomial representations of general linear groups. He was elected Fellow of the Royal Society of Edinburgh (1968) and Fellow of the Royal Society of London (1987). He was awarded prizes of the London Mathematical Society, a Senior Berwick Prize (in 1984) and the De Morgan Medal (in 2001). In his doctoral thesis, on semigroups, Sandy introduced fundamental relations, now known as ‘Green's relations’. He determined the characters of arbitrary finite general linear groups published 1955. Sandy then turned to representations of finite groups over fields of prime characteristic; his work laid the foundations for the module theoretic approach to the subject. His next highlight is his monograph on polynomial representations of GL n , published in 1980, which has become the basis for algebraic highest weight theory. Furthermore, in 1995 he proved a fundamental result on Hall algebras, establishing a connection between quantum groups and representations of finite-dimensional quiver algebras.

Download Full-text

Young modules for symmetric groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700002846 ◽

2001 ◽

Vol 71 (2) ◽

pp. 201-210 ◽

Cited By ~ 23

Author(s):

Karin Erdmann

Keyword(s):

Representation Theory ◽

Finite Groups ◽

Symmetric Groups ◽

General Linear ◽

Linear Groups ◽

Schur Algebras ◽

General Linear Groups ◽

Green Correspondence

AbstractLet K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.

Download Full-text

Estimating deep Littlewood-Richardson Coefficients

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2403 ◽

2014 ◽

Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽

Author(s):

Hariharan Narayanan

Keyword(s):

Representation Theory ◽

Approximation Scheme ◽

General Linear ◽

Linear Groups ◽

General Linear Groups ◽

International Audience ◽

Structure Constants ◽

Strongly Polynomial

International audience Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups $(GL_n)$. The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for Littlewood-Richardson coefficients corresponding to indices sufficiently far from the boundary of the Littlewood Richardson cone. 2. A proof of approximate log-concavity of the above mentioned class of Littlewood-Richardson coefficients. Coefficients de Littlewood Richardson sont des constantes de structure apparaissant dans la théorie de la représentation des groupes linéaires généraux $(GL_n)$. Les principaux résultats de cette étude sont les suivants: 1. Un schéma d’approximation polynomiale randomisée fortement pour des coefficients de Littlewood-Richardson correspondant aux indices suffisamment loin de la limite du cône Littlewood Richardson. 2. Une preuve de l’approximatif log-concavité de la classe de coefficients de Littlewood-Richardson mentionné ci-dessus.

Download Full-text