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Tensor Products of Representations of Finite Groups

Tensor Products of Representations of Finite Groups TENSOR PRODUCTS OF REPRESENTATIONS OF FINITE GROUPS R. M. BRYAN T AND L. G. KOVACS 1. Throughout this note K will be a field and G a finite group. By a KG-module we shall mean a finite-dimensional right module for the group algebra KG. A well- known theorem states that if V is a KG-module which is faithful for G then each (r) irreducible KG-module is a composition factor of some tensor power V of V. (See Burnside [2; §226], Fong and Gaschiitz [3], Steinberg [5] and Brauer [1; Theorem 1* and Remarks 4, 5].) Here we shall give an extremely simple proof of the somewhat stronger fact: each principal indecomposable KG-module is isomorphic to a direct summand of some tensor power of V. Our main result is THEOREM 1. If, for each g e G\l , V is a KG-module on which g has non-scalar action, then the regular KG-module is isomorphic to a direct summand of Here, and below, we have the convention that any empty tensor product of KG-modules is equal to the trivial one-dimensional KG-module T. By the regular KG-module we mean KG itself, regarded as a KG-module; and an element g of G is http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Tensor Products of Representations of Finite Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/4.2.133
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Abstract

TENSOR PRODUCTS OF REPRESENTATIONS OF FINITE GROUPS R. M. BRYAN T AND L. G. KOVACS 1. Throughout this note K will be a field and G a finite group. By a KG-module we shall mean a finite-dimensional right module for the group algebra KG. A well- known theorem states that if V is a KG-module which is faithful for G then each (r) irreducible KG-module is a composition factor of some tensor power V of V. (See Burnside [2; §226], Fong and Gaschiitz [3], Steinberg [5] and Brauer [1; Theorem 1* and Remarks 4, 5].) Here we shall give an extremely simple proof of the somewhat stronger fact: each principal indecomposable KG-module is isomorphic to a direct summand of some tensor power of V. Our main result is THEOREM 1. If, for each g e G\l , V is a KG-module on which g has non-scalar action, then the regular KG-module is isomorphic to a direct summand of Here, and below, we have the convention that any empty tensor product of KG-modules is equal to the trivial one-dimensional KG-module T. By the regular KG-module we mean KG itself, regarded as a KG-module; and an element g of G is

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1972

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