In the theory of cluster algebra, a mutation loop is a certain equivalence class of a sequence of seed mutations and permutations of indices. They form a group called the cluster modular group, which can be regarded as a combinatorial generalization of the mapping class groups of marked surfaces.

We introduce a new property of mutation loops which we call the “sign stability”, as a generalization of the pseudo-Anosov property of a mapping class. A sign-stable mutation loop has a numerical invariant which we call the “cluster stretch factor”, in analogy with the stretch factor of a pA mapping class. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two are estimated by the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

(This seminar was held on zoom.)