Odd and even major indices and one-dimensional characters for classical Weyl groups

TL;DR

This paper introduces odd and even versions of the major index for classical Weyl groups, demonstrating explicit factorization of their generating functions twisted by characters, extending several classical identities and results.

Contribution

It develops new odd and even major index statistics for Weyl groups and provides explicit factorizations of their generating functions, extending classical combinatorial identities.

Findings

Explicit factorization of generating functions for odd and even major indices

Extension of Carlitz's identity and Gessel-Simion Theorem

Refined parabolic extension of Wachs' result

Abstract

We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz's identity, of the Gessel-Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.