Complexity of the usual torus action on Kazhdan-Lusztig varieties

2021/11/26 by Donten-Bury, Maria, Escobar, Laura, Portakal, Irem · 3 citations
#05C20 (Secondary) #05E10 (Primary) #14M15 #14M25 #52B20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.13540

Abstract

We investigate the class of Kazhdan-Lusztig varieties, and its subclass of matrix Schubert varieties, endowed with a naturally defined torus action. Writing a matrix Schubert variety X_w as X_w=Y_w× ℂ^d (where d is maximal possible), we show that Y_w can be of complexity-k exactly when k≠ 1. Also, we give a combinatorial description of the extremal rays of the weight cone of a Kazhdan-Lusztig variety, which in particular turns out to be the edge cone of an acyclic directed graph. As a consequence we show that given permutations v and w, the complexity of Kazhdan-Lusztig variety indexed by (v,w) is the same as the complexity of the Richardson variety indexed by (v,w). Finally, we use this description to compute the complexity of certain Kazhdan-Lusztig varieties.

Cited by

• Cyclomatic numbers and permutations
• Torus Actions on Matrix Schubert and Kazhdan-Lusztig Varieties, and their Links to Statistical Models
• Complexity of the Zero Set of a Matrix Schubert Ideal

Related

• Torus Actions on Matrix Schubert and Kazhdan-Lusztig Varieties, and their Links to Statistical Models
• Complexity of the Zero Set of a Matrix Schubert Ideal
• Orbit structures and complexity in Schubert and Richardson Varieties
• A Generalization of Deodhar's Framework for Questions in Kazhdan-Lusztig Theory
• Toric matrix Schubert varieties and their polytopes