On the complete cd-index of a Bruhat interval

TL;DR

This paper investigates the non-negativity of the complete cd-index for Bruhat intervals, linking combinatorial path constructions and shelling conditions to prove non-negativity for certain monomials.

Contribution

It constructs path sets corresponding to cd-monomials and establishes a flip condition that ensures non-negativity, extending known results to broader cases.

Findings

Coefficients for monomials with at most one d are non-negative.

The flip condition relates to shelling of Bruhat intervals.

Path constructions provide combinatorial interpretations of coefficients.

Abstract

We study the non-negativity conjecture of the complete cd-index of a Bruhat interval defined by Billera and Brenti. For each cd-monomial M we construct a set of paths, such that if a "flip condition" is satisfied, then the number of these paths is the coefficient of the monomial M in the complete cd-index. When the monomial contains at most one d, then the condition follows from Dyer's proof of Cellini's conjecture. Hence the coefficients of these monomials are non-negative. We also relate the flip condition to shelling of Bruhat intervals.