2024/09/30 by Toshiya Yurikusa, Yurikusa, Toshiya
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2410.00133
For the cluster algebra A associated with a triangulated surface, we give a characterization of the triangulated surface such that different non-initial cluster monomials in A have different f-vectors. Similarly, for the associated Jacobian algebra J, we give a characterization of the triangulated surface such that different τ-rigid J-modules have different dimension vectors. Moreover, we also show that different basic support τ-tilting J-modules have different dimension vectors. Our main ingredient is a notion of intersection numbers defined by Qiu and Zhou. As an application, we show that the denominator conjecture holds for A if the marked surface is a closed surface with exactly one puncture, or the given tagged triangulation has neither loops nor tagged arcs connecting punctures.