A conjecture on descents, inversions and the weak order

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 $B$).

💡 Research Summary

The paper introduces a novel notion of “partition” for elements of an arbitrary Coxeter system ((W,S)). For a given element (w\in W), its left inversion set (T(w)={t\in T\mid \ell(tw)<\ell(w)}) is required to be the disjoint union of the left inversion sets of a collection (\mathcal P\subseteq W). When (|\mathcal P|=2) the authors call ({u,v}) a bipartition of (w). The central conjecture (Conjecture 1) states that whenever ({u,v}) is a bipartition of (w) the number of right descents satisfies
\