Frieze varieties are invariant under Coxeter mutation
Abstract
We define a generalized version of the frieze variety introduced by Lee, Li, Mills, Seceleanu and the second author. The generalized frieze variety is an algebraic variety determined by an acyclic quiver and a generic specialization of cluster variables in the cluster algebra for this quiver. The original frieze variety is obtained when this specialization is (1, . . . , 1). The main result is that a generalized frieze variety is determined by any generic element of any component of that variety. We also show that the "Coxeter mutation" cyclically permutes these components. In particular, this shows that the frieze variety is invariant under the Coxeter mutation at a generic point. The paper contains many examples which are generated using a new technique which we call an invariant Laurent polynomial. We show that a symmetry of a mutation of a quiver gives such an invariant rational function.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2019
- DOI:
- 10.48550/arXiv.1906.00106
- arXiv:
- arXiv:1906.00106
- Bibcode:
- 2019arXiv190600106I
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Algebraic Geometry;
- 13F60;
- 16G20
- E-Print:
- 13 pages. v2: Key observations by Gordana Todorov added. Presented by the first author and referenced by the second author at the conference "Cluster Algebras and Representation Theory" held in Kyoto in June 2019