$0$-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra

Abstract

We define an action of the $0$-Hecke algebra of type A 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. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics.

Keywords

• $0$-hecke algebra
• stanley-reisner ring
• boolean algebra
• noncommutative hall-littlewood symmetric function
• multivariate quasisymmetric function.
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
• [math.math-co] mathematics [math]/combinatorics [math.co]