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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An optimization problem for unitary and orthogonal representations of finite groups

Authors: D. Ž. Djoković and I. F. Blake
Journal: Trans. Amer. Math. Soc. 164 (1972), 267-274
MSC: Primary 20.80
MathSciNet review: 0285629
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Abstract: Let $ G \to {\text{GL}}(V)$ be a faithful orthogonal representation of a finite group G acting in an Euclidean space V. For a unit vector x we choose $ g \ne 1$ in G so that $ \vert gx - x\vert$ is minimal and put $ \delta (x) = \vert gx - x\vert$. We study the class of vectors x which maximize $ \delta (x)$ and have the additional property that $ \vert gx - x\vert$ depends only on the conjugacy class of $ g \in G$. For some special types of representations we are able to characterize completely this class of vectors.

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Keywords: Unitary and orthogonal representations, characters, irreducible modules, homogeneous modules, group algebra, Schur's lemma, Wedderburn's theorem
Article copyright: © Copyright 1972 American Mathematical Society

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