Integer Representations of the Generalized Symmetric Groups

2022/11/25 by Hasan Arslan, Arslan, Hasan, Alnour Altoum +3
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2211.13961

Abstract

In this paper, we construct a mixed-base number system over the generalized symmetric group G(m,1,n), which is a complex reflection group with a root system of type B_n^(m). We also establish one-to-one correspondence between all positive integers in the set \1,⋯,mⁿn!\ and the elements of G(m,1,n) by constructing the subexceedant function in relation to this group. In addition, we provide a new enumeration system for G(m,1,n) by defining the inversion statistic on G(m,1,n). Finally, we prove that the flag-major index is equi-distributed with this inversion statistic on G(m,1,n). Therefore, the flag-major index is Mahonian on G(m,1,n) with respect to the length function L.

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