Group-theoretical generalization of necklace polynomials

Let G be a group, U a subgroup of G of finite index, X a finite alphabet and q an indeterminate. In this paper, we study symmetric polynomials MG(X,U) and M q G(X,U) which were introduced as a group-theoretical generalization of necklace polynomials. Main results are to generalize identities satisfied by necklace polynomials due to Metropolis and Rota in a bijective way, and to express M G(X,U) in terms of MG(X,V )’s, where [V ] ranges over a set of conjugacy classes of subgroups to which U is subconjugate. As a byproduct, we provide the explicit form of the GLm(C)-module whose character is M q Z (X,nZ), where m is the cardinality of X.