Intersection cohomology of the symmetric reciprocal plane

TL;DR

This paper calculates the Kazhdan-Lusztig polynomial for a specific uniform matroid and links its coefficients to combinatorial chord arrangements, revealing connections to symmetric group representations.

Contribution

It provides a combinatorial formula for the polynomial coefficients and identifies the intersection cohomology groups with specific symmetric group representations.

Findings

Coefficients correspond to non-intersecting chord arrangements in polygons

Intersection cohomology groups match irreducible symmetric group representations

Explicit formula for Kazhdan-Lusztig polynomial of the uniform matroid

Abstract

We compute the Kazhdan-Lusztig polynomial of the uniform matroid of rank n-1 on n elements by proving that the i-th coefficient of is equal to the number of ways to choose i non-intersecting chords in an (n-i+1)-gon. We also show that the corresponding intersection cohomology group is isomorphic to the irreducible representation of the symmetric group associated to the partition [n-2i,2,...,2].