scholarly journals Young modules for symmetric groups

2001 ◽  
Vol 71 (2) ◽  
pp. 201-210 ◽  
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.

Semisimple Schur Algebras

1998 ◽  
Vol 124 (1) ◽  
pp. 15-20 ◽  
Author(s):  
STEPHEN R. DOTY ◽  
DANIEL K. NAKANO
Keyword(s):  
Representation Theory ◽  
General Linear ◽  
Linear Groups ◽  
Infinite Field ◽  
Polynomial Representation ◽  
Schur Algebras ◽  
Symmetric Powers ◽  
General Linear Groups ◽  
Finite Dimensional ◽  
Frobenius Kernels

Schur algebras are certain finite-dimensional algebras that completely control the polynomial representation theory of the general linear groups over an infinite field. Infinitesimal Schur algebras are truncated versions of the classical Schur algebras which control the polynomial representation theory of the Frobenius kernels of general linear groups. In this paper we use some elementary results on symmetric powers to classify the semisimple Schur algebras. We then classify the semisimple infinitesimal Schur algebras as well.

scholarly journals MODULAR DECOMPOSITION NUMBERS OF CYCLOTOMIC HECKE AND DIAGRAMMATIC CHEREDNIK ALGEBRAS: A PATH THEORETIC APPROACH

2018 ◽  
Vol 6 ◽  
Author(s):  
C. BOWMAN ◽  
A. G. COX
Keyword(s):  
Representation Theory ◽  
Symmetric Groups ◽  
Upper Bounds ◽  
Linear Groups ◽  
Theoretic Approach ◽  
Strong Linkage ◽  
Cherednik Algebras ◽  
Arbitrary Characteristic ◽  
General Linear Groups ◽  
Decomposition Numbers

We introduce a path theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher-level generalizations over fields of arbitrary characteristic. Our first main result is a ‘super-strong linkage principle’ which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalize the notion of homomorphisms between Weyl/Specht modules which are ‘generically’ placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.

scholarly journals The generalized Gelfand–Graev characters of GLn(Fq)

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.

scholarly journals Estimating deep Littlewood-Richardson Coefficients

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.

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