Group-theoretical generalization of necklace polynomials

초록

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) 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) in terms of M-G(X, V)'s, where left perpendicularVright perpendicular 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-Z(q) (X, nZ), where m is the cardinality of X.

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