0-Hecke algebra actions on coinvariants and flags

TL;DR

This paper explores the action of the 0-Hecke algebra on coinvariant algebras and flag varieties, revealing new algebraic and combinatorial interpretations of permutation statistics and symmetric functions.

Contribution

It demonstrates that the coinvariant algebra admits the regular representation of the 0-Hecke algebra, extending classical results to this algebraic setting.

Findings

Coinvariant algebra carries the regular representation of H_n(0).

Generated functions for permutations are interpreted via H_n(0) actions.

Action on Springer fibers links to Hall-Littlewood symmetric functions.

Abstract

The 0-Hecke algebra $H_n(0)$ is a deformation of the group algebra of the symmetric group $\SS_n$. We show that its coinvariant algebra naturally carries the regular representation of $H_n(0)$, giving an analogue of the well-known result for $\SS_n$ by Chevalley-Shephard-Todd. By investigating the action of $H_n(0)$ on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of $H_n(0)$ on the cohomology rings of the Springer fibers, and similarly interpret the (noncommutative) Hall-Littlewood symmetric functions indexed by hook shapes.