Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.

Assem, Ibrahim; Reutenauer, Christophe

doi:10.5802/ambp.304

Mutating seeds: types

𝔸

and

\tilde{𝔸}

Assem, Ibrahim ¹ ; Reutenauer, Christophe ²

¹ Département de mathématiques, Université de Sherbrooke Sherbrooke (Québec) J1K2R1,Canada
² Laboratoire de combinatoire et d’informatique mathématique, Université du Québec à Montréal Case postale 8888, succursale Centre-ville Montréal (Québec) H3C 3P8, Canada

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 29-73.

Résumé

In the cases $𝔸$ and $\tilde{𝔸}$ , we describe the seeds obtained by sequences of mutations from an initial seed. In the $\tilde{𝔸}$ case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.

MR Zbl

DOI : 10.5802/ambp.304

Classification : 13F60, 16G20, 16G99
Mots-clés : Cluster algebras, mutations, seeds, quivers

Affiliations des auteurs :

Assem, Ibrahim ¹ ; Reutenauer, Christophe ²

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     title = {Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.},
     journal = {Annales math\'ematiques Blaise Pascal},
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Assem, Ibrahim; Reutenauer, Christophe. Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.. Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 29-73. doi : 10.5802/ambp.304. http://www.numdam.org/articles/10.5802/ambp.304/

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