A Cluster Expansion Formula ($A_n$ case)
Copyright policy )A Cluster Expansion Formula ($A_n$ case)
We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\Sigma$ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $\Sigma$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $\Sigma$.
- University of Massachusetts Amherst United States
Ptolemy cluster algebra, cluster variable, 16S99 (primary), Cluster algebras, regular polygon, Associative rings and algebras arising under various constructions, triangulation, Mathematics - Rings and Algebras, Combinatorial aspects of groups and algebras, 16S99 (primary); 5E15; 16G20 (secondary), Rings and Algebras (math.RA), 5E15, FOS: Mathematics, Representations of quivers and partially ordered sets, mutation, 16G20 (secondary), Representation Theory (math.RT), Mathematics - Representation Theory, cluster algebra
Ptolemy cluster algebra, cluster variable, 16S99 (primary), Cluster algebras, regular polygon, Associative rings and algebras arising under various constructions, triangulation, Mathematics - Rings and Algebras, Combinatorial aspects of groups and algebras, 16S99 (primary); 5E15; 16G20 (secondary), Rings and Algebras (math.RA), 5E15, FOS: Mathematics, Representations of quivers and partially ordered sets, mutation, 16G20 (secondary), Representation Theory (math.RT), Mathematics - Representation Theory, cluster algebra
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