Distinguished filtrations of the $0$-Hecke modules for dual immaculate quasisymmetric functions

Abstract

Let $\alpha$ range over the set of compositions. Dual immaculate quasisymmetric functions ${\mathfrak{S}}_{\alpha }^{\ast }$, introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki, provide a quasisymmetric analogue of Schur functions. They also constructed an indecomposable $0$-Hecke module ${\mathcal{V}}_{\alpha }$ whose image under the quasisymmetric characteristic is ${\mathfrak{S}}_{\alpha }^{\ast }$. In this paper, we prove that ${\mathcal{V}}_{\alpha }$ admits a distinguished filtration with respect to the basis of Young quasisymmetric Schur functions. This result offers a novel representation-theoretic interpretation of the positive expansion of ${\mathfrak{S}}_{\alpha }^{\ast }$ in the basis of Young quasisymmetric Schur functions. A key tool in our proof is Mason's analogue of the Robinson-Schensted-Knuth algorithm, for which we establish a version of Green's theorem. As an unexpected byproduct of our investigation, we construct an indecomposable $0$-Hecke module ${\mathbf{Y}}_{\alpha }$ whose image under the quasisymmetric characteristic is the Young quasisymmetric Schur function ${\hat{\mathcal{S}}}_{\alpha }$. Further properties of this module are also investigated. And, by applying a suitable automorphism twist to this module, we obtain an indecomposable $0$-Hecke module whose image under the quasisymmetric characteristic is the quasisymmetric Schur function ${\mathcal{S}}_{\alpha }$.

50 pages