Abstract
Abstract
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.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis