Distinguished filtrations of the $0$-Hecke modules for dual immaculate quasisymmetric functions
So-Yeon Lee, Young-Tak Oh
TL;DR
This work develops a representation-theoretic framework for dual immaculate and extended quasisymmetric functions by analyzing distinguished filtrations of associated indecomposable $0$-Hecke modules. Central to the approach is Mason’s analogue of the Robinson–Schensted–Knuth algorithm, complemented by a Greene-type theorem, which together connect tableau combinatorics with left Bruhat interval modules and filtrations relative to bases $\mathcal{S}$ and $\hat{\mathcal{S}}$. The authors prove that $\mathcal{V}_\alpha$ admits a distinguished filtration with respect to $\hat{\mathcal{S}}$ for all $\alpha$, and, in the shuffle case, $X_\alpha$ does as well; they construct indecomposable $H_n(0)$-modules $\mathbf{Y}_\alpha$ with $\mathrm{ch}([\mathbf{Y}_\alpha]) = \hat{\mathscr{S}}_\alpha$ and obtain a $\upphi$-twist giving $\mathscr{S}_\alpha$. A key outcome is a surjection ladder from projective indecomposables to $\mathcal{V}_\alpha$, $X_\alpha$, $\mathbf{Y}_\alpha$, and $\hat{\mathbf{S}}_{\alpha,C}$, linking positive $F$-expansions to concrete module filtrations. The results provide a conceptual bridge between the combinatorics of Young composition tableaux and the representation theory of $0$-Hecke algebras, with implications for indecomposability questions and future structural classifications.
Abstract
Let $α$ range over the set of compositions. Dual immaculate quasisymmetric functions $\mathfrak{S}_α^*$, introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki, provide a quasisymmetric analogue of Schur functions. They also constructed an indecomposable $0$-Hecke module $\mathcal{V}_α$ whose image under the quasisymmetric characteristic is $\mathfrak{S}_α^*$. In this paper, we prove that $\mathcal{V}_α$ 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 $\mathfrak{S}_α^*$ 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 $\mathbf{Y}_α$ whose image under the quasisymmetric characteristic is the Young quasisymmetric Schur function $\hat{\mathscr{S}}_α$. 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 $\mathscr{S}_α$.