Complexity of the usual torus action on Kazhdan-Lusztig varieties

TL;DR

This paper studies the torus action on Kazhdan-Lusztig varieties, revealing their complexity properties, providing combinatorial descriptions of their weight cones, and relating their complexity to that of Richardson varieties.

Contribution

It characterizes the complexity of matrix Schubert varieties, describes the extremal rays of their weight cones combinatorially, and relates Kazhdan-Lusztig variety complexity to Richardson varieties.

Findings

Y_w can be of complexity-k exactly when k≠1.

The extremal rays of the weight cone form the edge cone of an acyclic directed graph.

Complexity of Kazhdan-Lusztig varieties equals that of corresponding Richardson varieties.

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 $\overline{X_w}$ as $\overline{X_w}=Y_w\times \mathbb{C}^d$ (where $d$ is maximal possible), we show that $Y_w$ can be of complexity-$k$ exactly when $k\neq 1$. 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.