Separable elements and splittings of Weyl groups

TL;DR

This paper studies separable elements in finite Weyl groups, generalizing separable permutations, and explores their role in group splittings, combinatorial inequalities, and geometric structures like graph associahedra.

Contribution

It extends the concept of separable permutations to Weyl groups, classifies group splittings via order ideals generated by separable elements, and links these to combinatorial and geometric structures.

Findings

Multiplication map W/U x U -> W is length-additive and bijective under certain conditions.

Classifies quotients of symmetric groups that induce splittings, solving a longstanding problem.

Provides a combinatorial proof of Sidorenko's inequality and introduces a new q-analog.

Abstract

We continue the study of separable elements in finite Weyl groups. These elements generalize the well-studied class of separable permutations. We show that the multiplication map $W/U \times U \to W$ is a length-additive bijection, or splitting, of the Weyl group $W$ when $U$ is an order ideal in right weak order generated by a separable element; this generalizes a result for the symmetric group, answering an open problem of Wei. For a generalized quotient of the symmetric group, we show that this multiplication map is a bijection if and only if $U$ is an order ideal in right weak order generated by a separable element, thereby classifying those generalized quotients which induce splittings of the symmetric group, resolving a problem of Bj\"{o}rner and Wachs from 1988. We also prove that this map is always surjective when $U$ is an order ideal in right weak order. Interpreting these sets of permutations as linear extensions of 2-dimensional posets gives the first direct combinatorial proof of an inequality due originally to Sidorenko in 1991, answering an open problem Morales, Pak, and Panova. We also prove a new $q$-analog of Sidorenko's formula. All of these results are conjectured to extend to arbitrary finite Weyl groups. Finally, we show that separable elements in $W$ are in bijection with the faces of all dimensions of several copies of the graph associahedron of the Dynkin diagram of $W$. This correspondence associates to each separable element $w$ a certain nested set; we give product formulas for the rank generating functions of the principal upper and lower order ideals generated by $w$ in terms of these nested sets, generalizing several known formulas.