Modules of the $0$-Hecke algebra arising from standard permuted composition tableaux

We study the $H_n(0)$-module $\mathbf{S}^\sigma_\alpha$ due to Tewari and van Willigenburg, which was constructed using new combinatorial objects called standard permuted composition tableaux and decomposed into cyclic submodules. First, we show that every direct summand appearing in their decomposition is indecomposable and characterize when $\mathbf{S}^\sigma_\alpha$ is indecomposable. Second, we find characteristic relations among $\mathbf{S}^\sigma_\alpha$'s and expand the image of $\mathbf{S}^\sigma_\alpha$ under the quasi characteristic in terms of quasisymmetric Schur functions. Finally, we show that the canonical submodule of $\mathbf{S}^\sigma_\alpha$ appears as a homomorphic image of a projective indecomposable module.