Signed Mahonian on Parabolic Quotients of Colored Permutation Groups

TL;DR

This paper extends the signed Mahonian polynomial to colored permutation groups and their parabolic quotients, providing new formulas and generalizations of previous results in permutation group theory.

Contribution

It introduces a generalized signed Mahonian polynomial for colored permutation groups and their quotients, expanding on prior work with new formulas and subgroup analyses.

Findings

Derived the signed Mahonian polynomial for parabolic quotients of colored permutation groups.

Generalized Biagioli's result to even signed permutation groups.

Provided product formulas for these polynomials.

Abstract

We study the generating polynomial of the flag major index with each one-dimensional character, called signed Mahonian polynomial, over the colored permutation group, the wreath product of a cyclic group with the symmetric group. Using the insertion lemma of Han and Haglund-Loehr-Remmel and a signed extension established by Eu et al., we derive the signed Mahonian polynomial over the quotients of parabolic subgroups of the colored permutation group, for a variety of systems of coset representatives in terms of subsequence restrictions. This generalizes the related work over parabolic quotients of the symmetric group due to Caselli as well as to Eu et al. As a byproduct, we derive a product formula that generalizes Biagioli's result about the signed Mahonian on the even signed permutation groups.