A strong connection between cluster algebras and representation theory was established by the cluster category. Cluster characters, like the original Caldero-Chapoton (CC) map, are maps from certain triangulated categories to cluster algebras and they have generated much interest. Holm and Jørgensen constructed a modified CC map from a sufficiently nice triangulated category to a commutative ring, which is a generalised frieze under some conditions. In their construction, a quotient $K_{0}^{sp}(\mathcal{T})/M$ of a Grothendieck group of a cluster tilting subcategory $\mathcal{T}$ is used. In this article, we show that this quotient is the Grothendieck group of a certain extriangulated category, thereby exposing the significance of it and the relevance of extriangulated structures. We use this to define another modified CC map that recovers the one of Holm--Jørgensen.
We prove our results in a higher homological context. Suppose $\mathcal{S}$ is a $(d+2)$-angulated category with subcategories $\mathcal{X}\subseteq\mathcal{T}\subseteq\mathcal{S}$, where $\mathcal{X}$ is functorially finite and $\mathcal{T}$ is $2d$-cluster tilting, satisfying some mild conditions. We show there is an isomorphism between the Grothendieck group $K_{0}(\mathcal{S},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$ of the category $\mathcal{S}$, equipped with the $d$-exangulated structure induced by $\mathcal{X}$, and the quotient $K_{0}^{sp}(\mathcal{T})/N$, where $N$ is the higher analogue of $M$ above. When $\mathcal{X}=\mathcal{T}$ the isomorphism is induced by the higher index with respect to $\mathcal{T}$ introduced recently by Jørgensen. Thus, in the general case, we can understand the map taking an object in $\mathcal{S}$ to its $K_{0}$-class in $K_{0}(\mathcal{S},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$ as a higher index with respect to the rigid subcategory $\mathcal{X}$.
A geometric realization of socle-projective categories for posets of type $\mathbb{A}$
本文建立了一个链接的理论之间的集群代数和理论的表示偏序集。我们引入一类偏序集，通过要求避免某些类型的峰值subposets，并表明这些偏序集可以实现为具有某些额外的箭头的$\mathbb{A}$型箭图的偏序集。这类偏序集因此被称为\emph{posets of type $\mathbb{A}$}。然后，我们给出了$\mathbb{A}$型偏序集的关联代数上的可生成柱脚投射模范畴作为正多边形的某些对角线的组合范畴的几何实现。这种构造受到Caldero、Chapoton和第一作者将$\mathbb{A}$型的簇范畴实现为所有对角线的范畴的启发。我们还研究了在上述构造下，对应于socle-投射的簇变量所生成的簇代数的子代数。给出了这个子代数与整个簇代数相等的一个充分条件。
