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arXiv · 2309.02932

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

Ming Liu·Houyi Yu
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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 $B$.

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Ming Liu, Houyi Yu. 2023-09-06. Separable elements and splittings in Weyl groups of Type $B$. https://arxiv.org/abs/2309.02932

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