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arXiv · 2412.09227

A conjecture on descents, inversions and the weak order

Christophe Hohlweg·Viviane Pons
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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 $B$).

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Christophe Hohlweg, Viviane Pons. 2026-03-12. A conjecture on descents, inversions and the weak order. https://arxiv.org/abs/2412.09227

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