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Generalised Cogrowth series, random walks, and the group determinant

Published online by Cambridge University Press:  14 August 2017

STEPHEN P. HUMPHRIES
STEPHEN P. HUMPHRIES*
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A. e-mail: [email protected]
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Abstract

We associate to a group G a series that generalises the cogrowth series of G and is related to a random walk on G. We show that the series is rational if and only if G is finite, generalizing a result of Kouksov [Kou]. We show that when G is finite, the series determines G. There are naturally occurring ideals and varieties that are acted on by Aut(G). We study these and generalize this to the context of S-rings over finite groups. There is an associated representation of Aut(G) and we characterize when this is irreducible.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 445 - 465
Copyright
Copyright © Cambridge Philosophical Society 2017 

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