Character Sums for Cayley Graphs

Abstract

Following [1], by a Cayley digraph we mean a graph Cay(G, S) whose vertex set is a group G, and there exists a directed edge from a vertex g to another vertex h if g −1 h ∈ S, where S is a generating subset of G. The graph Cay(G, S) is called a Cayley graph if S = S −1 and 1 ∉ S. In Problem 3.3 of the above cited article, the following question is proposed. Let G be a finite group, let Γ = Cay(G, S) be a Cayley digraph, ν a positive integer, and where χ1, …, χ h are all irreducible characters of G. Is the set M ν = {μ i | χ i (1) = ν} an invariant of Γ? (Thus, does Cay(G, S) ≅ Cay(G, S′) imply ?) It is easy to see that the set M ν is an invariant for the Cayley digraphs if the underlying group G is abelian. Here we negatively answer the above problem. We show that for every n ≥ 4 there is a Cayley graph Γ n on the symmetric group S n so that the above set M ν is not an invariant of Γ n . We also find some other groups with the latter property.