# Separable elements in Weyl groups

## TL;DR

This paper introduces the concept of separable elements in Weyl groups, demonstrating their structural properties and pattern avoidance characterization, and resolving an open problem related to rank-generating functions.

## Contribution

It generalizes separable permutations to Weyl groups and proves their order ideals are rank-symmetric, rank-unimodal, and satisfy a specific product formula.

## Key findings

- Order ideals generated by separable elements are rank-symmetric and rank-unimodal.
- Product of rank generating functions of these ideals equals that of the entire group.
- Separable elements are characterized by pattern avoidance.

## Abstract

We define the notion of a separable element in a finite Weyl group, generalizing the well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are rank-symmetric and rank-unimodal, and that the product of their rank generating functions gives that of the whole group, answering an open problem of Fan Wei. We also prove that separable elements are characterized by pattern avoidance in the sense of Billey and Postnikov.