A characterization of Kazhdan-Lusztig right cells containing smooth elements

TL;DR

This paper classifies Kazhdan-Lusztig right cells in the symmetric group that contain smooth elements related to smooth Schubert varieties, providing conditions and algorithms to identify these cells and their smooth elements.

Contribution

It offers a complete classification of KL right cells with smooth elements, along with conditions and an algorithm for identifying smooth elements within these cells.

Findings

Complete classification of KL right cells with only smooth elements

Sufficient conditions for cells to contain only non-smooth or some smooth elements

An efficient algorithm to identify smooth elements in a given cell

Abstract

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$. Its Weyl group is the symmetric group $S_n$. In this paper, we want to describe some Kazhdan-Lusztig right cells containing smooth elements which parameterize the smooth Schubert varieties. These elements are closely related to the study of associated varieties of highest weight modules of $\mathfrak{sl}(n,\mathbb{C})$. Firstly, we give a complete classification of the KL right cells containing only smooth elements. Then we give a sufficient condition for a KL right cell to contain only non-smooth elements by using invariant subsequences and a sufficient condition for a KL right cell to contain some smooth elements. Finally, we give an efficient algorithm to find out all the smooth elements in a given KL right cell.