Three-term recurrence relations of minimal affinizations of type $G_2$
Abstract
Minimal affinizations form a class of modules of quantum affine algebras introduced by Chari. We introduce a system of equations satisfied by the $q$-characters of minimal affinizations of type $G_2$ which we call the M-system of type $G_2$. The M-system of type $G_2$ contains all minimal affinizations of type $G_2$ and only contains minimal affinizations. The equations in the M-system of type $G_2$ are three-term recurrence relations. The M-system of type $G_2$ is much simpler than the extended T-system of type $G_2$ obtained by Mukhin and the second author. We also interpret the three-term recurrence relations in the M-system of type $G_2$ as exchange relations in a cluster algebra constructed by Hernandez and Leclerc.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2014
- DOI:
- 10.48550/arXiv.1412.3884
- arXiv:
- arXiv:1412.3884
- Bibcode:
- 2014arXiv1412.3884Q
- Keywords:
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- Mathematics - Quantum Algebra;
- 17B37
- E-Print:
- 23 pages. The original name of the paper is: Cluster algebras and minimal affinizations of representations of the quantum group of type $G_2$