\(\mathcal{G}\)-systems (Q2217549)

\textit{Cluster algebras}, introduced by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] in early 2000, are algebras with a kind of combinatorial structure based on transformations called \textit{mutations} between sets called \textit{clusters}. A cluster is a set of Laurent polynomials called \textit{cluster variables}, and mutation is the operation of replacing one of these cluster variables with another cluster variable to give a new cluster. Nowadays, this combinatorial structure is known as a universal structure found in various fields in mathematics such as representation theory of algebras, hyperbolic geometry, number theory, etc. In this paper, we focus on a family of matrices called the (family of) \(G\)-matrices, which appear in cluster algebra theory. This family of matrices was first introduced by \textit{S. Fomin} and \textit{A. Zelevinsky} [Compos. Math. 143, No. 1, 112--164 (2007; Zbl 1127.16023)] in 2007 as the matrices associated with clusters in cluster algebra, consisting of the grading vectors (\(g\)-vectors) of the cluster variables. Later, \textit{R. Dehy} and \textit{B. Keller} [Int. Math. Res. Not. 2008, Article ID rnn029, 17 p. (2008; Zbl 1144.18009)] introduced an analogous concept into the representation theory of algebra, and \textit{R. Schiffler} [Adv. Math. 223, No. 6, 1885--1923 (2010; Zbl 1238.13029)] introduced an analogous concept into marked surfaces. These matrices play an important role in each theory, and their properties and applications are still being clarified by many researchers. The main theme of this paper is to pick out the essential properties common to the families of \(G\)-matrices, and to axiomatize of \(G\)-matrices to treat them in a unified framework. For example, group theory was developed based on the idea of treating the structure of product operations found in various places as ``groups'' and investigating its properties. This paper proposes to use the same idea to understand \(G\)-matrices that appear in various fields unifiedly. In the paper, we redefine a \(G\)-matrix as an element of a family of matrices satisfying three conditions called the \textit{mutation condition}, the \textit{Bongartz co-completion condition}, and the \textit{Uniqueness condition}, and call a family of matrices satisfying these conditions a \textit{\(\mathcal G\)-system}. In addition, the following two things about the \(\mathcal G\)-system are shown. \begin{itemize} \item[1.]properties of \(G\)-matrices derived from the axioms of the \(\mathcal {G}\)-system, \item[2.]the fact that the families of \(G\)-matrices given by previous studies in various fields satisfy the axioms of the \(\mathcal G\)-system. \end{itemize} 1. In the context of cluster algebra, the question of whether row vectors of \(G\)-matrices are sign-coherent or not was an open problem for a long time until it was solved by \textit{M. Gross} et al. [J. Am. Math. Soc. 31, No. 2, 497--608 (2018; Zbl 1446.13015)] in 2019. In the paper, we derive the sign-coherence of the row vectors of the \(G\)-matrix from the axioms of the \(\mathcal{G}\)-system without using the properties of any particular field. In addition, some properties such as the compatibility of mutation and Bongartz co-completion, which have been proved independently in different fields, are proved using the axioms of the \(\mathcal{G}\)-system. 2. In the paper, we construct \(\mathcal {G}\)-systems that appear in cluster algebra theory, \(\tau\)-tilting theory, silting theory, cluster tilting theory, and unpunctured surfaces, and prove that it satisfies the axioms. In this process, it is given the operations of Bongartz co-completion in cluster algebra theory and unpunctured surfaces.