Geodesic
Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions
Forum of Mathematics, Sigma, Tome 10 (2022)

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Let n be a nonnegative integer. For each composition $\alpha $ of n, Berg, Bergeron, Saliola, Serrano and Zabrocki 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 $\mathrm {Ext}^1_{H_n(0)}(\mathcal {V}_{\alpha }, \mathbf {F}_{\beta })$ and $\mathrm {Ext}^1_{H_n(0)}( \mathbf {F}_{\beta }, \mathcal {V}_{\alpha })$, where $\mathbf {F}_{\beta }$ is the simple $H_n(0)$-module attached to a composition $\beta $ of n. We also compute $\mathrm {Ext}_{H_n(0)}^i(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ when $i=0,1$ and $\beta \le _l \alpha $, where $\le _l$ represents the lexicographic order on compositions. 
@article{10_1017_fms_2022_81,
     author = {Seung-Il Choi and Young-Hun Kim and Sun-Young Nam and Young-Tak Oh},
     title = {Homological properties of {0-Hecke} modules for dual immaculate quasisymmetric functions},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {10},
     year = {2022},
     doi = {10.1017/fms.2022.81},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.81/}
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Seung-Il Choi; Young-Hun Kim; Sun-Young Nam; Young-Tak Oh. Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.81

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