Cluster algebra: Difference between revisions

Suppose '''S''' is a compact connected oriented Riemann surface and '''M''' is a non-empty finite set of points in '''S''' that contains at least one point from each boundary component of '''S''' (the boundary of '''S''' is not assumed to be either empty or non-empty). The pair ('''S''','''M''') is often referred to as a ''bordered surface with makedmarked points''. It has been shown by Fomin-Shapiro-Thurston that if '''S''' is not a closed surface, or if '''M''' has more than one point, then the (tagged) arcs on ('''S''','''M''') parameterize the set of cluster variables of certain cluster algebra ''A''('''S''','''M'''), which depends only on ('''S''','''M''') and the choice of some coefficient system, in such a way that the set of (tagged) triangulations of ('''S''','''M''') is in one-to-one correspondence with the set of clusters of ''A''('''S''','''M'''), two (tagged) triangulations being related by a ''flip'' if and only if the clusters they correspond to are related by cluster mutation.