The projective cover of tableau-cyclic indecomposable $H_n(0)$-modules

Abstract:

Let $\alpha$ be a composition of $n$ and $\sigma$ a permutation in $\mathfrak {S}_{\ell (\alpha )}$. This paper concerns the projective covers of $H_n(0)$-modules $\mathcal {V}_\alpha$, $X_\alpha$, and $\mathbf {S}^\sigma _{\alpha }$ whose images under the quasisymmetric characteristic are the dual immaculate quasisymmetric function, the extended Schur function, and the quasisymmetric Schur function when $\sigma$ is the identity, respectively. First, we show that the projective cover of $\mathcal {V}_\alpha$ is the projective indecomposable module $\mathbf {P}_\alpha$ due to Norton, and $X_\alpha$ and the $\phi$-twist of the canonical submodule $\mathbf {S}^{\sigma }_{\beta ,C}$ of $\mathbf {S}^\sigma _{\beta }$ for $(\beta ,\sigma )$’s satisfying suitable conditions appear as homomorphic images of $\mathcal {V}_\alpha$. Second, we introduce a combinatorial model for the $\phi$-twist of $\mathbf {S}^\sigma _{\alpha }$ and derive a series of surjections starting from $\mathbf {P}_\alpha$ to the $\phi$-twist of $\mathbf {S}^\mathrm {id}_{\alpha ,C}$. Finally, we construct the projective cover of every indecomposable direct summand $\mathbf {S}^\sigma _{\alpha , E}$ of $\mathbf {S}^\sigma _{\alpha }$. As a byproduct, we give a characterization of triples $(\sigma ,\alpha ,E)$ such that the projective cover of $\mathbf {S}^\sigma _{\alpha ,E}$ is indecomposable.