Determinantal formulas with major indices

TL;DR

This paper provides a straightforward proof of a major index determinant formula for symmetric groups and extends the approach to signed and colored permutation groups, confirming several conjectures.

Contribution

It introduces a simple proof technique for the major index determinant formula and generalizes it to other permutation groups, confirming existing conjectures.

Findings

Proof of the major index determinant formula for symmetric groups

Extension of the formula to signed and colored permutation groups

Confirmation of Krattenthaler's conjectures for these groups

Abstract

We give a simple proof of a major index determinant formula in the symmetric group discovered by Krattenthaler and first proved by Thibon using noncommutative symmetric functions. We do so by proving a factorization of an element in the group ring of the symmetric group. By applying similar methods to the groups of signed permutations and colored permutations, we prove determinant formulas in these groups as conjectured by Krattenthaler.