2016/11/24 by Jan Geuenich, Daniel Labardini-Fragoso, Geuenich, Jan +1 · 2 citations
Mathematics · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.1611.08301
Let Σ=(\Σ,M,O) be a surface with marked points and\norder-2 orbifold points which is either unpunctured or once-punctured closed,\nand \ω:O\→ 1,4 a function. For each triangulation \τ of\n Σ we construct a cochain complex C^ bullet(\τ,\ω). A\ncolored triangulation is defined to be a pair consisting of a triangulation\n\τ and a 1-cocycle of C^ bullet(\τ,\ω); the combinatorial notion of\ncolored flip of colored triangulations is then defined as a refinement of the\nnotion of flip of triangulations. Our main construction associates to each\ncolored triangulation a species and a potential, and our main result shows that\ncolored triangulations related by a colored flip have SPs related by the\ncorresponding SP-mutation.\n We define the flip graph of (\Σ,M,O,\ω), whose vertices are the\npairs (\τ,x) with \τ a triangulation and x a cohomology class in\nH1(C^ bullet(\τ,\ω)), with an edge between (\τ,x) and (\σ,z)\niff (\τ,\ξ) and (\σ,\ζ) are related by a colored flip for some\ncocycles \ξ and \ζ respectively representing x and z. We prove that\nthis graph is disconnected if \Σ is not contractible.\n For unpunctured surfaces we show that (\τ,\ξ) and (\τ,\ξ') yield\nisomorphic Jacobian algebras if and only if [\ξ]=[\ξ'] in cohomology. We\nprove that every SP-realization of any (\τ,\ω) via a non-degenerate SP\nover a cyclic Galois extension with certain roots of unity is right-equivalent\nto one of the SPs we construct here.\n The species constructed here are species realizations of the 2|O|\nskew-symmetrizable matrices assigned by Felikson-Shapiro-Tumarkin to any given\n\τ. In the prequel to this paper we realized only one of these matrices via\nspecies, but therein we allowed the presence of arbitrarily many punctures.\n