0-Hecke algebras of the symmetric groups: centers and modules associated to quasisymmetric Schur functions

Abstract

We consider two aspects of 0-Hecke algebras of symmetric groups: their centers and modules associated to quasisymmetric Schur functions. Tewari and van Willigenburg constructed 0-Hecke modules that are mapped to the skew quasisymmetric Schur functions by the quasisymmetric characteristic. These include straight modules that correspond to the ordinary quasisymmetric Schur functions of Haglund, Luoto, Mason and van Willigenburg. The modules admit a natural direct sum decomposition. We study the summands and provide combinatorial rules for their tops and socles. Moreover, we show that they are indecomposable in the straight case. This is a difference to the general skew case where the summands can be decomposable. For a certain kind of skew modules, we describe a decomposition into indecomposable projective submodules. Vector space bases of the centers of the 0-Hecke algebras of the symmetric groups were described by He. These bases are indexed by certain equivalence classes of permutations whose explicit description is rather complicated. Even their number is not obvious. Building on work of Geck, Kim and Pfeiffer we obtain a set of representatives. This leads to a parametrization of the equivalence classes by certain compositions called maximal. Moreover, we develop an explicit combinatorial description for the equivalence classes indexed by maximal compositions whose odd parts form a hook. We infer that except for the identity the elements of He’s basis corresponding to these equivalence classes annihilate all simple 0-Hecke modules belonging to the nontrivial block of their 0-Hecke algebra.