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 M _G (X,U) and M_G^q(X,U)M_{G}^{q}(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^q(X,U)M_{G}^{q}(X,U) in terms of M _G (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 GL _m (ℂ)-module whose character is M_\mathbbZ^q(X,n\mathbbZ)M_{\mathbb{Z}}^{q}(X,n\mathbb{Z}), where m is the cardinality of X.