Pentagonal Relations and the Exchange Module of the Type $A_n$ Cluster Algebra

Mathematics – Combinatorics

Scientific paper

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10 pages, 9 figures

Scientific paper

In this paper we study relations between the exchange relations in the cluster algebra of type-$A_n$, which has the 1-skeleton of the associahedron as its exchange graph, and the complex of triangulations of a regular $(n+3)$-gon, $\TN$, as its cluster complex. We define the exchange module of the type-$A_n$ cluster algebra to be the $\mathbb{Z}$-module generated by all differences of exchangeable cluster variables. Using a notion of discrete homotopy theory, we describe all of the relations in the exchange module and show that these relations are generated by $\binom{n+2}{4}$ five term relations which correspond to equivalence classes in the abelianization of the discrete fundamental group of $\TN$. Finally, we show that the exchange module is a free $\mathbb{Z}$-module and give a minimal generating set of size $\binom{n+2}{3}$.

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