An Inversion Statistic on the Hyperoctahedral Group

TL;DR

This paper introduces a new inversion statistic for the hyperoctahedral group, explores its properties, and demonstrates its equidistribution with the flag-major index, providing a new combinatorial framework for understanding $B_n$.

Contribution

It defines a novel inversion statistic on $B_n$, establishes its properties, and shows its equidistribution with the flag-major index, enhancing combinatorial analysis of reflection groups.

Findings

Inversion statistic on $B_n$ is combinatorially characterized.

An enumeration system for $B_n$ is developed.

The inversion statistic is shown to be equidistributed with the flag-major index.

Abstract

In this paper, we introduce an inversion statistic on the hyperoctahedral group $B_n$ by using an decomposition of a positive root system of this reflection group. Then we prove some combinatorial properties for the inversion statistic. We establish an enumeration system on the group $B_n$ and give an efficient method to uniquely derive any group element known its enumeration order with the help of the inversion table. In addition, we prove that the \textit{flag-major index} is equi-distributed with this inversion statistic on $B_n$.