Odd length in Weyl groups

TL;DR

This paper introduces the odd length statistic for Weyl groups, generalizes existing types, and demonstrates that their generating functions factor nicely across most types, revealing new algebraic structures.

Contribution

It defines a new odd length statistic on Weyl groups, extending previous definitions, and proves factorization properties of their generating functions across various types.

Findings

Generating functions factor nicely in most Weyl group types

Multivariate analogues of factorizations are obtained for types B and D

Potential exceptions in type E8 are noted

Abstract

We define a new statistic on any Weyl group which we call the odd length and which reduces, for Weyl groups of types $A$, $B$, and $D$, the the statistics by the same name that have already been defined and studied in [10], [13], [14], and [3]. We show that the signed (by length) generating function of the odd length always factors nicely except possibly in type $E_8$, and we obtain multivariate analogues of these factorizations in types $B$ and $D$.