New graphs of finite mutation type

Harm Derksen ; Theodore Owen

The electronic journal of combinatorics, Tome 15 (2008)

Résumé

To a directed graph without loops or $2$-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph $\Gamma$ is the set of all isomorphism classes of graphs that can be obtained from $\Gamma$ by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with marked points. We will call such graphs "of geometric type". Besides graphs with $2$ vertices, and graphs of geometric type, there are only 9 other "exceptional" mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes.

Détail

DOI : 10.37236/863

Classification : 05C20, 05C76, 05E99
Mots-clés : directed graph, skew symmetric matrix, mutations of skew symmetric matrices, mutations of skew symmetrizable matrices, mutation class, mutation finite graph, graphs of geometric type, exceptional mutation classes, finite mutation classes