0-Hecke algebra actions on flags, polynomials, and Stanley-Reisner rings

Abstract

We study combinatorial aspects of the representation theory of the 0-Hecke algebra $H_n(0)$, a deformation of the group algebra of the symmetric group $\SS_n$. We study the action of $H_n(0)$ on the polynomial ring in $n$ variables. We show that the coinvariant algebra of this action naturally carries the regular representation of $H_n(0)$, giving an analogue of the well-known result for the symmetric group by Chevalley-Shephard-Todd. By investigating the action of $H_n(0)$ on coinvariants and %finite 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 $H_n(0)$-action on the cohomology rings of the Springer fibers, and similarly interpret the (noncommutative) Hall-Littlewood symmetric functions indexed by hook shapes.We generalize the last result from hooks to all compositions by defining an $H_n(0)$-action on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki, and to a more general family of noncommutative symmetric functions having parameters associated with paths in binary trees introduced recently by Lascoux, Novelli, and Thibon. We also obtain multivariate quasisymmetric function identities from this $H_n(0)$-action, which specialize to results of Garsia and Gessel on generating functions of multivariate distributions of permutation statistics. More generally, for any finite Coxeter group $W$, we define an action of its Hecke algebra $H_W(q)$ on the Stanley-Reisner ring of its Coxeter complex. We find the invariant algebra of this action, and show that the coinvariant algebra of this action is isomorphic to the regular representation of $H_W(q)$ if $q$ is generic. When $q=0$ we find a decomposition for the coinvariant algebra as a multigraded $H_W(0)$-module.