Integer Representations of the Generalized Symmetric Groups

TL;DR

This paper introduces a new mixed-base number system for the generalized symmetric group G(m,1,n), establishing a bijection with integers, defining a new enumeration method, and proving the equidistribution of the flag-major index and inversion statistic.

Contribution

It constructs a novel mixed-base representation and enumeration system for G(m,1,n), linking combinatorial statistics with group elements.

Findings

Established a bijection between integers and group elements.

Defined a new inversion statistic for G(m,1,n).

Proved the flag-major index is Mahonian on G(m,1,n).

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 $\{1,\cdots,m^nn!\}$ 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 \textit{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 $L$.