Separable elements and splittings in Weyl groups of Type $B$
Ming Liu, Houyi Yu
TL;DR
The paper extends the Gaetz–Gao framework from symmetric groups to Weyl groups of type $\mathrm{B}_n$ by classifying separable and minimal non-separable signed permutations via forbidden patterns and establishing a precise equivalence: for $U=[e,u]_R$ in a type $\mathrm{B}$ Weyl group $W$, the splitting $W/U$ with $U$ exists if and only if $u$ is separable. The authors develop a root-system pattern avoidance description in $\mathfrak{B}_n$, analyze minimal non-separable elements and their inverses, and prove that non-separable $u$ fail to yield a splitting, while separable $u$ do yield one, extending the conjecture to type $B$. They also examine rank-symmetry properties of lower ideals and provide constructive and inductive arguments to support the splitting results. Overall, the work advances understanding of generalized quotients, weak-order structure, and splitting phenomena in Weyl groups, with potential implications for representations and combinatorial pattern avoidance in non-symmetric settings.
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 $B$.