Abstract
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 [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 \(M_{\mathbb{Z}}^{q}(X,n\mathbb{Z})\), where m is the cardinality of X.
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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0015281).
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Oh, YT. Group-theoretical generalization of necklace polynomials. J Algebr Comb 35, 389–420 (2012). https://doi.org/10.1007/s10801-011-0307-3
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DOI: https://doi.org/10.1007/s10801-011-0307-3