Schur algebras and 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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On decomposition numbers and Alvis–Curtis duality

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000667 ◽

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.

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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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