We study the $H_n(0)$-module ${\mathbf{S}}_{\alpha }^{\sigma }$ 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 ${\mathbf{S}}_{\alpha }^{\sigma }$ is indecomposable. Second, we find characteristic relations among ${\mathbf{S}}_{\alpha }^{\sigma }$'s and expand the image of ${\mathbf{S}}_{\alpha }^{\sigma }$ under the quasi characteristic in terms of quasisymmetric Schur functions. Finally, we show that the canonical submodule of ${\mathbf{S}}_{\alpha }^{\sigma }$ appears as a homomorphic image of a projective indecomposable module.

中文翻译：

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标准排列组合表产生的0-Hecke代数的模

我们研究 $H_{ñ}（0）$-模块 ${\mathbf{小号}}_{\alpha }^{\sigma }$由于Tewari和van Willigenburg，他们是使用称为标准置换组合表格的新组合对象构建的，并分解为循环子模块。首先，我们证明了分解中出现的每个直接求和都是不可分解的，并描述了何时${\mathbf{小号}}_{\alpha }^{\sigma }$是不可分解的。其次，我们发现${\mathbf{小号}}_{\alpha }^{\sigma }$并扩大形象 ${\mathbf{小号}}_{\alpha }^{\sigma }$在准对称Schur函数方面具有准特性。最后，我们证明了的规范子模块${\mathbf{小号}}_{\alpha }^{\sigma }$ 显示为射影不可分解模块的同态图像。