Configuration spaces, \(\mathrm{FS}^{\mathrm{op}}\)-modules, and Kazhdan-Lusztig polynomials of braid matroids. (English) Zbl 06747482

Proudfoot, Nicholas; Young, Ben

New York J. Math. 23, 813-832 (2017).

Summary: 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 \(\mathbb C\), regarded as a graded representation of the symmetric group \(S_n\). We show that, in fixed cohomological degree, this sequence of representations of symmetric groups naturally admits the structure of an \(\mathrm{FS}\)-module, and that the dual \(\mathrm{FS}^{\mathrm{op}}\)-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.

MSC:

20C30	Representations of finite symmetric groups
55R80	Discriminantal varieties and configuration spaces in algebraic topology
55N33	Intersection homology and cohomology in algebraic topology

Keywords:

configuration space; representation stability; Kazhdan-Lusztig polynomial