Configuration spaces, $\operatorname{FS^{op}}$-modules, and Kazhdan-Lusztig polynomials of braid matroids

TL;DR

This paper explores the structure of equivariant Kazhdan-Lusztig polynomials of braid matroids, revealing their interpretation as symmetric group representations and establishing their FS-module structure for asymptotic analysis.

Contribution

It demonstrates that these representations form finitely generated FS-modules and applies existing work to derive asymptotic formulas and restrictions on irreducible components.

Findings

Representations form finitely generated FS-modules.

Asymptotic formulas for representation dimensions.

Restrictions on irreducible representation components.

Abstract

The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in the plane, regarded as a graded representation of the symmetric group. We show that, in fixed cohomological degree, this sequence of representations of symmetric groups naturally admits the structure of an FS-module, and that the dual FS^op-module is finitely generated. Using the work of Sam and Snowden, we give an asymptotic formula for the dimensions of these representations and obtain restrictions on which irreducible representations can appear in their decomposition.