A conjecture on descents, inversions and the weak order

TL;DR

This paper explores a conjecture relating descents and partitions of elements in Coxeter systems, proving it for symmetric and hyperoctahedral groups, with implications in algebraic geometry and combinatorics.

Contribution

It introduces a new conjecture linking descents of elements to their partitions in Coxeter systems and proves it for specific types.

Findings

Conjecture holds for symmetric groups (type A)

Conjecture holds for hyperoctahedral groups (type B)

Links partitions of elements to descents in Coxeter systems

Abstract

In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal P\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the left inversion set of the elements in $\mathcal P$. Partitions of elements of $W$ arises in the study of the Belkale-Kumar product on the cohomology $H^*(X,\mathbb Z)$, where $X$ is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group $\mathcal S_n$ are also related to the {\em Babington-Smith model} in algebraic statistics or to the simplicial faces of the Littlewood-Richardson cone. We state the conjecture that the number of right descents of $w$ is the sum of the number of right descents of the elements of $\mathcal P$ and prove that this conjecture holds in the cases of symmetric groups (type $A$) and hyperoctahedral groups (type $B$).