The cluster complex for cluster Poisson varieties and representations of acyclic quivers

2023/10/05 by Melo, Carolina, Chávez, Alfredo Nájera
#13F60 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2310.03626

Abstract

Let X be a skew-symmetrizable cluster Poisson variety. The cluster complex Δ⁺(X) was introduced by Gross, Hacking, Keel and Kontsevich. It codifies the theta functions on X that restrict to a character of a seed torus. Every seed \bf s for X determines a fan realization Δ⁺_{\bf s}(X) of Δ⁺(X). For every \bf s we provide a simple and explicit description of the cones of Δ⁺_{\bf s}(X) and their facets using \bf c-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of Δ⁺ _{\bf s}(X) in terms of F-polynomials. In case X is skew-symmetric and the quiver Q associated to \bf s is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of Δ⁺_{\bf s}(X) using \bf g-vectors of (non-necessarily rigid) objects in K^{\rm b}(proj kQ).

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