Parabolically induced functions and equidistributed pairs

TL;DR

The paper introduces a general method to extend functions equidistributed with length from parabolic subgroups to entire Coxeter groups, applicable to various known combinatorial statistics across different types.

Contribution

It provides a novel procedure to construct length-equidistributed functions on entire Coxeter groups from those on parabolic subgroups, generalizing previous results.

Findings

The method applies to known functions like major index and negative major index.

It works across all finite Coxeter groups.

The approach is valid in the broader context of graded posets.

Abstract

Given a function defined over a parabolic subgroup of a Coxeter group, equidistributed with the length, we give a procedure to construct a function over the entire group, equidistributed with the length. Such a procedure permits to define functions equidistributed with the length in all the finite Coxeter groups. We can establish our results in the general setting of graded posets which satisfy some properties. These results apply to some known functions arising in Coxeter groups as the major index, the negative major index and the D-negative major index defined in type $A$, $B$ and $D$ respectively.