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 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.

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.

scholarly journals On the Homology of GLn and Higher Pre-Bloch Groups

2000 ◽  
Vol 52 (6) ◽  
pp. 1310-1338 ◽  
Author(s):  
Serge Yagunov
Keyword(s):  
Spectral Sequence ◽  
Higher Dimensions ◽  
Natural Generalization ◽  
General Linear ◽  
Linear Groups ◽  
Spectral Sequences ◽  
Infinite Field ◽  
General Linear Groups

AbstractFor every integer n > 1 and infinite field F we construct a spectral sequence converging to the homology of GLn(F) relative to the group of monomial matrices GMn(F). Some entries in E2-terms of these spectral sequences may be interpreted as a natural generalization of the Bloch group to higher dimensions. These groups may be characterized as homology of GLn relatively to GLn-1 and GMn. We apply the machinery developed to the investigation of stabilization maps in homology of General Linear Groups.

scholarly journals On decomposition numbers and Alvis–Curtis duality

2007 ◽  
Vol 143 (3) ◽  
pp. 509-520
Author(s):  
BERND ACKERMANN ◽  
SIBYLLE SCHROLL
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
Schur Algebras ◽  
General Linear Groups ◽  
Decomposition Numbers

AbstractWe show that for general linear groups GLn(q) as well as for q-Schur algebras the knowledge of the modular Alvis–Curtis duality over fields of characteristic ℓ, ℓ ∤ q, is equivalent to the knowledge of the decomposition numbers.

scholarly journals general linear groups

1997 ◽  
Vol 90 (3) ◽  
pp. 549-576 ◽  
Author(s):  
Avner Ash ◽  
Mark McConnell
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
General Linear Groups
Sign in / Sign up

Export Citation Format

Share Document