Some representation theory for the modular general linear groups

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.

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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.

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Semisimple Schur Algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004197002466 ◽

1998 ◽

Vol 124 (1) ◽

pp. 15-20 ◽

Cited By ~ 14

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.

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