2018/03/22 by Kyungyong Lee, Li Li, Lee, Kyungyong +7
Mathematics · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.1803.08459
We introduce a new class of algebraic varieties which we call frieze\nvarieties. Each frieze variety is determined by an acyclic quiver. The frieze\nvariety is defined in an elementary recursive way by constructing a set of\npoints in affine space. From a more conceptual viewpoint, the coordinates of\nthese points are specializations of cluster variables in the cluster algebra\nassociated to the quiver.\n We give a new characterization of the finite--tame--wild trichotomy for\nacyclic quivers in terms of their frieze varieties. We show that an acyclic\nquiver is representation finite, tame, or wild, respectively, if and only if\nthe dimension of its frieze variety is 0,1, or \≥2, respectively.\n