$F$-matrices of cluster algebras from triangulated surfaces

Abstract

For a given marked surface $(S,M)$ and a fixed tagged triangulation $T$ of $(S,M)$, we show that each tagged triangulation $T^{\prime }$ of $(S,M)$ is uniquely determined by the intersection numbers of tagged arcs of $T$ and tagged arcs of $T^{\prime }$. As consequence, each cluster in the cluster algebra $\mathcal{A}(T)$ is uniquely determined by its $F$-matrix which is a new numerical invariant of the cluster introduced by Fujiwara and Gyoda.

33 pages

Cited by in corpus (3)

• The valuation pairing on an upper cluster algebra
• Relation between $f$-vectors and $d$-vectors in cluster algebras of finite type or rank 2
• Compatibility degree of cluster complexes