Separable elements and splittings in Weyl groups of Type $B$

TL;DR

This paper classifies separable elements in Weyl groups of type B, confirms a conjecture about splittings related to separable elements, and characterizes minimal non-separable permutations through forbidden patterns.

Contribution

It extends the classification of separable elements and the splitting conjecture from symmetric groups to Weyl groups of type B, providing a comprehensive understanding of their structure.

Findings

Classified all separable and minimal non-separable signed permutations.

Confirmed the Gaetz-Gao conjecture for Weyl groups of type B.

Characterized separable elements via forbidden patterns.

Abstract

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair $(X,Y)$ of subsets of the symmetric group $\mathfrak{S}_n$, the multiplication map $X\times Y\rightarrow \mathfrak{S}_n$ is a splitting (i.e., a length-additive bijection) of $\mathfrak{S}_n$ if and only if $X$ is the generalized quotient of $Y$ and $Y$ is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type $B$.