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

TL;DR

This paper introduces a 0-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra, leading to new multivariate noncommutative symmetric functions and identities that connect to existing symmetric function theories.

Contribution

It defines a novel algebraic action and constructs a family of multivariate noncommutative symmetric functions that generalize known functions and identities.

Findings

Derived multivariate noncommutative symmetric functions from the algebra action.

Specialized these functions to known noncommutative Hall-Littlewood and (q,t)-analogues.

Established new identities for multivariate permutation statistics.

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, 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, which specialize to results of Garsia and Gessel on generating functions of multivariate distributions of permutation statistics.