Sign-twisted generating functions of odd length over any parabolic quotients of the even hyperoctahedral groups

Haihang Gu, Houyi Yu

Published 2023-03-14, updated 2023-06-11Version 2

The odd length on Weyl groups is a new statistic analogous to the classical Coxeter length function, and features combinatorial and parity conditions. We establish an explicit closed product formula of the sign-twisted generating functions of the odd length for any parabolic quotients of the even hyperoctahedral groups (Weyl groups of type $D$). As a consequence, we verify three conjectures posed by Brenti and Carnevale. We then give necessary and sufficient conditions for the sign-twisted generating functions to be not expressible as products of cyclotomic polynomials. It turns out that there is at most one non-cyclotomic factor, which always has the form $x^n + 2x^m + 1$ for some positive integers $m$ and $n$, settling a conjecture of Stembridge.