Flip Posets of Bruhat Intervals

Saúl A Blanco

doi:10.37236/7910

Abstract

In this paper we introduce a way of partitioning the paths of shortest lengths in the Bruhat graph $B(u,v)$ of a Bruhat interval $[u,v]$ into rank posets $P_{i}$ in a way that each $P_{i}$ has a unique maximal chain that is rising under a reflection order. In the case where each $P_{i}$ has rank three, the construction yields a combinatorial description of some terms of the complete $\textbf{cd}$-index as a sum of ordinary $\textbf{cd}$-indices of Eulerian posets obtained from each of the $P_{i}$.

Highlights

Given a Coxeter group W with u, v ∈ W, the Bruhat graph associated with a Bruhat interval [u, v] has u-v paths of several lengths
When applied to the u-v paths of length two and three, the corresponding terms of the complete cd-index can be obtained as the sum of the cd-index of certain Eulerian posets determined by the algorithm
We show that certain terms of the complete cd-index, those corresponding to u-v paths of length two or three, can be expressed as a sum of the cd-index of certain Eulerian posets that are obtained from the Flip algorithm, enabling us to conclude nonnegativity in these cases, as well as a connection between the two indices

Summary

Introduction

Given a Coxeter group W with u, v ∈ W , the Bruhat graph associated with a Bruhat interval [u, v] has u-v paths of several lengths (see [9]). When applied to the u-v paths of length two and three, the corresponding terms of the complete cd-index can be obtained as the sum of the cd-index of certain Eulerian posets determined by the algorithm. We show that certain terms of the complete cd-index, those corresponding to u-v paths of length two or three, can be expressed as a sum of the cd-index of certain Eulerian posets that are obtained from the Flip algorithm, enabling us to conclude nonnegativity in these cases, as well as a connection between the two indices

Basic definitions

Flip algorithm

Identification

Terms of degree two in the complete cd-index