Séminaire Lotharingien de Combinatoire, 93B.55 (2025), 12 pp.
Ziyao Fu, Ni Tang, Sylvester W. Zhang and Yunchi Zheng
Expansion Formulae for SL3 Fock-Goncharov Cluster Algebras
Abstract.
We investigate the cluster algebra of Fock--Goncharov coordinates of the moduli space of decorated SL3-local systems ASL3,S when S is a marked disk. In particular, we give explicit expansion formula for the cluster variables with coefficients in terms of dimer covers (or perfect matchings) of certain subgraphs of Goncharov's A2 plabic graphs. We describe the poset structures on the said dimer covers, and show that the corresponding rank functions compute the F-polynomials of the corresponding cluster variables.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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