Identities for Poincar\'e polynomials via Kostant cascades

TL;DR

This paper establishes a new identity connecting the Poincaré polynomials of stabilizer subgroups in affine Weyl groups and their Weyl group counterparts, enhancing understanding of their algebraic structure.

Contribution

It introduces and proves a novel identity linking Poincaré polynomials of stabilizer subgroups in affine and finite Weyl groups.

Findings

Proved a new identity relating affine and finite Weyl group stabilizers.

Enhanced understanding of the algebraic structure of Weyl groups.

Provided tools for further research in representation theory and algebraic combinatorics.

Abstract

We propose and prove an identity relating the Poincar\'e polynomials of stabilizer subgroups of the affine Weyl group and of the corresponding stabilizer subgroups of the Weyl group.