Algebraic entropy of sign-stable mutation loops

開催日時: 
2020/06/04 Thu 10:30 - 12:00
講演者: 
石橋典
講演者所属: 
京都大学数理解析研究所
概要: 

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.

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