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

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

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