Generalised Cogrowth series, random walks, and the group determinant
2017 ◽
Vol 165
(3)
◽
pp. 445-465
Keyword(s):
We associate to a groupGa series that generalises the cogrowth series ofGand is related to a random walk onG. We show that the series is rational if and only ifGis finite, generalizing a result of Kouksov [Kou]. We show that whenGis finite, the series determinesG. There are naturally occurring ideals and varieties that are acted on by Aut(G). We study these and generalize this to the context ofS-rings over finite groups. There is an associated representation of Aut(G) and we characterize when this is irreducible.