0-Hecke algebra actions on coinvariants and flags

Abstract

The 0-Hecke algebra H n (0) is a deformation of the group algebra of the symmetric group $\mathfrak{S}_{n}$ . We show that its coinvariant algebra naturally carries the regular representation of H n (0), giving an analogue of the well-known result for $\mathfrak{S}_{n}$ 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 (non-commutative) Hall–Littlewood symmetric functions indexed by hook shapes.