Vol. 78, No. 1, 1978

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On relations for representations of finite groups

Christopher Lloyd Morgan

Vol. 78 (1978), No. 1, 157–159
Abstract

Let G be a finite group, and suppose that

A : G → GL(n,C )

is a (complex) representation of G with character χ. A (complex) linear relation for A is a formal complex linear combination gGagg such that gGagA(g) = 0.

We prove the following theorem, which determines the linear relations in terms of the character χ.

Theorem. Let A be a representation for a finite group G, let χ be the character of A, and let {g1,,gk} be a subset of G. Then j=1kajgj is a relation for A if and only if j=1kχ(gigj1)aj = 0, for all i = 1,,k.

Mathematical Subject Classification 2000
Primary: 20C15
Milestones
Received: 30 June 1977
Revised: 21 January 1978
Published: 1 September 1978
Authors
Christopher Lloyd Morgan