对偶完美拟对称函数的0-Hecke模的同调性质
Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions
摘要：
设$n$为非负整数。Berg $\textit{et al.}$对$n$的每一个组合$α$引入了一个循环不可分解的$H_n(0)$ -模$\mathcal{V}_α$，它具有一个对偶完美拟对称函数作为拟对称特征的映象.本文从同调的角度研究了$\mathcal{V}_α$的性质。为了更准确地说，我们构造了$\mathcal{V}_α$的最小投射表示和$\mathcal{V}_α$的最小内射表示。使用它们，我们计算${\rm Ext}^1_{H_n(0)}(\mathcal{V}_α, {\bf F}_β)$和${\rm Ext}^1_{H_n(0)}( {\bf F}_β, \mathcal{V}_α)$，其中${\bf F}_β$是简单的$H_n(0)$ -模块附加到一个组合$n$的$β$。当$i=0,1$和$β\le_l α$同时存在时，我们也计算了${\rm Ext}_{H_n(0)}^i(\mathcal{V}_α,\mathcal{V}_β)$，其中$\le_l$表示组合的词典顺序。
英文摘要：
Let $n$ be a nonnegative integer. For each composition $α$ of $n$, Berg $\textit{et al.}$ introduced a cyclic indecomposable $H_n(0)$-module $\mathcal{V}_α$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal{V}_α$'s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal{V}_α$ and a minimal injective presentation of $\mathcal{V}_α$ as well. Using them, we compute ${\rm Ext}^1_{H_n(0)}(\mathcal{V}_α, {\bf F}_β)$ and ${\rm Ext}^1_{H_n(0)}( {\bf F}_β, \mathcal{V}_α)$, where ${\bf F}_β$ is the simple $H_n(0)$-module attached to a composition $β$ of $n$. We also compute ${\rm Ext}_{H_n(0)}^i(\mathcal{V}_α,\mathcal{V}_β)$ when $i=0,1$ and $β\le_l α$, where $\le_l$ represents the lexicographic order on compositions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
