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

Abstract

Let n be a nonnegative integer. For each composition α of n, Berg et al. introduced a cyclic indecomposable H n (0)-module V α with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study V α 's from the homological viewpoint. To be precise, we construct a minimal projective presentation of V α and a minimal injective presentation of V α as well. Using them, we compute Ext 1 Hn(0) (V α , F β ) and Ext 1 Hn(0) (F β , V α ), where F β is the simple H n (0)-module attached to a composition β of n. We also compute Ext i Hn(0) (V α , V β ) when i = 0, 1 and β ≤ l α, where ≤ l represents the lexicographic order on compositions.

Diagram supermodules for $0$-Hecke-Clifford algebras

“…There has been much recent activity concerning construction of 0-Hecke modules whose quasisymmetric characteristics are elements of noteworthy bases of QSym, e.g., [4,6,31,34,38,39], and also in determining the structure of these modules, e.g., [12,13,14,23,25]. It is therefore natural to ask the same question for 0-Hecke-Clifford algebras: can one construct 0-Hecke-Clifford supermodules whose peak characteristics are noteworthy families or bases of functions in Peak?…”