Singularities of Schubert Varieties within a Right Cell

Abstract

We describe an algorithm that pattern embeds, in the sense of Woo-Yong, any Bruhat interval of a symmetric group into an interval whose extremes lie in the same right Kazhdan-Lusztig cell. This apparently harmless fact has applications in finding examples of reducible associated varieties of 𝔰𝔩ₙ-highest weight modules, as well as in the study of W-graphs for symmetric groups, and in comparing various bases of irreducible representations of the symmetric group or its Hecke algebra. For example, we can systematically produce many negative answers to a question from the 1980s of Borho-Brylinski and Joseph, which had been settled by Williamson via computer calculations only in 2014.