The projective cover of tableau-cyclic indecomposable 𝐻_{𝑛}(0)-modules

Abstract

Let α \alpha be a composition of n n and σ \sigma a permutation in S ℓ ( α ) \mathfrak {S}_{\ell (\alpha )} . This paper concerns the projective covers of H n ( 0 ) H_n(0) -modules V α \mathcal {V}_\alpha , X α X_\alpha , and S α σ \mathbf {S}^\sigma _{\alpha } whose images under the quasisymmetric characteristic are the dual immaculate quasisymmetric function, the extended Schur function, and the quasisymmetric Schur function when σ \sigma is the identity, respectively. First, we show that the projective cover of V α \mathcal {V}_\alpha is the projective indecomposable module P α \mathbf {P}_\alpha due to Norton, and X α X_\alpha and the ϕ \phi -twist of the canonical submodule S β , C σ \mathbf {S}^{\sigma }_{\beta ,C} of S β σ \mathbf {S}^\sigma _{\beta } for ( β , σ ) (\beta ,\sigma ) ’s satisfying suitable conditions appear as homomorphic images of V α \mathcal {V}_\alpha . Second, we introduce a combinatorial model for the ϕ \phi -twist of S α σ \mathbf {S}^\sigma _{\alpha } and derive a series of surjections starting from P α \mathbf {P}_\alpha to the ϕ \phi -twist of S α , C i d \mathbf {S}^\mathrm {id}_{\alpha ,C} . Finally, we construct the projective cover of every indecomposable direct summand S α , E σ \mathbf {S}^\sigma _{\alpha , E} of S α σ \mathbf {S}^\sigma _{\alpha } . As a byproduct, we give a characterization of triples ( σ , α , E ) (\sigma ,\alpha ,E) such that the projective cover of S α , E σ \mathbf {S}^\sigma _{\alpha ,E} is indecomposable.