Odd length for even hyperoctahedral groups and signed generating functions

TL;DR

The paper introduces a new statistic for even hyperoctahedral groups, computes its signed generating function, and demonstrates its factorization properties, extending previous work on Coxeter groups of types A and B.

Contribution

It defines a novel statistic on even hyperoctahedral groups, computes its signed generating function, and explores its algebraic factorization, advancing understanding of these groups.

Findings

Signed generating function factors nicely over the entire group

The statistic generalizes the odd length concept to even hyperoctahedral groups

Conjectures about further properties are proposed

Abstract

We define a new statistic on the even hyperoctahedral groups which is a natural analogue of the odd length statistic recently defined and studied on Coxeter groups of types $A$ and $B$. We compute the signed (by length) generating function of this statistic over the whole group and over its maximal and some other quotients and show that it always factors nicely. We also present some conjectures.