Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions

Abstract: Let $n$ be a nonnegative integer. For each composition $\alpha$ of $n$, Berg $\textit{et al.}$ introduced a cyclic indecomposable $H_n(0)$-module $\mathcal{V}\alpha$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal{V}\alpha$'s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal{V}\alpha$ and a minimal injective presentation of $\mathcal{V}\alpha$ as well. Using them, we compute ${\rm Ext}^1_{H_n(0)}(\mathcal{V}_\alpha, {\bf F}\beta)$ and ${\rm Ext}^1{H_n(0)}( {\bf F}\beta, \mathcal{V}\alpha)$, where ${\bf F}\beta$ is the simple $H_n(0)$-module attached to a composition $\beta$ of $n$. We also compute ${\rm Ext}{H_n(0)}^i(\mathcal{V}\alpha,\mathcal{V}{\beta})$ when $i=0,1$ and $\beta \le_l \alpha$, where $\le_l$ represents the lexicographic order on compositions.