发表机构
Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入簇代数上的G-分次种子概念，统一了符号种子、簇自同构群和分次簇代数，并应用于矩阵空间、positroid簇等例子，与热带突变建立联系。
AI 中文摘要
簇代数理论与全正性理论紧密相连；事实上，更好地理解全正性正是Fomin和Zelevinsky引入簇代数的主要动机之一[arXiv:math/0104151]。特别地，任何坐标环具有簇结构的簇簇都具有自然的正部分概念：簇中所有簇变量均为正值的子集。在本文中，我们解释了簇簇中还存在其他符号胞腔，从簇理论的角度来看它们同样自然。这些胞腔来源于符号种子，符号种子可视为簇变量上的Z/2Z分次，并在[arXiv:2310.17727]中于amplituhedron的背景下被引入。更一般地，给定任意阿贝尔群G，我们为簇代数引入G-分次种子的概念，这是一种将G的元素分配给每个簇变量且与簇结构相容的方式。当G为乘法群{-1,1}时，这恢复了上述符号种子的概念；当G=C*时，这恢复了簇自同构群[GSV10]或簇膨胀群[arXiv:2603.17890]的概念；当G=Z^d时，这恢复了Grabowski-Launois[arXiv:1301.2133]、Grabowski[arXiv:1309.6170]和Gekhtman-Shapiro-Vainshtein[GSV10,第5.2节]研究的分次簇代数概念（该概念此前在Fomin-Zelevinsky[arXiv:math/0602259]的工作中作为特殊情况出现）。我们研究的例子包括方阵空间、对称矩阵、斜对称矩阵、positroid簇和amplituhedron瓦片。我们还将这一概念与G=R或Z时的热带突变联系起来。
英文摘要
The theory of cluster algebras is closely connected to the theory of total positivity; indeed, the desire to better understand total positivity was one of the main motivations for Fomin and Zelevinsky's introduction of cluster algebras [arXiv:math/0104151]. In particular, any cluster variety whose coordinate ring has a cluster structure has a natural notion of positive part: the subset of the variety where all cluster variables are positive. In this paper, we explain that there are other signed cells contained in cluster varieties that are equally natural from a cluster-theoretic point of view. These come from signed seeds, which can be thought of as a $\mathbb{Z}/2\mathbb{Z}$-grading on cluster variables, and which were introduced in [arXiv:2310.17727] in the context of the amplituhedron. More generally, given any abelian group $G$, we introduce the notion of a $G$-graded seed for a cluster algebra, which is a way of assigning elements of $G$ to each cluster variable which is compatible with the cluster structure. When $G$ is the multiplicative group $\{-1, 1\}$, this recovers the above notion of signed seed; when $G = \mathbb{C}^*$, this recovers the notion of cluster automorphism group [GSV10] or cluster dilation group [arXiv:2603.17890]; and when $G = \mathbb{Z}^d$, this recovers the notion of graded cluster algebra studied by Grabowski-Launois [arXiv:1301.2133], Grabowski [arXiv:1309.6170] and Gekhtman-Shapiro-Vainshtein [GSV10, Section 5.2] (which had previously appeared in special cases in work of Fomin-Zelevinsky [arXiv:math/0602259]). The examples we study include the space of square matrices, symmetric matrices, skew-symmetric matrices, positroid varieties, and amplituhedron tiles. We also connect this notion to tropical mutation when $G = \mathbb{R}$ or $\mathbb{Z}$.
Comments47 pages, 14 figures
