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Exponents Associated with $Y$-Systems and their Relationship with $q$-Series
Yuma Mizuno Department of Mathematical and Computing Science, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
Abstract:
Let $X_r$ be a finite type Dynkin diagram, and $\ell$ be a positive integer greater than or equal to two. The $Y$-system of type $X_r$ with level $\ell$ is a system of algebraic relations, whose solutions have been proved to have periodicity. For any pair $(X_r, \ell)$, we define an integer sequence called exponents using formulation of the $Y$-system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type $X_r$, and prove this conjecture for $(A_1,\ell)$ and $(A_r, 2)$ cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from $q$-series identities for this relationship.
Keywords:
cluster algebras, $Y$-systems, root systems, $q$-series.
Received: September 27, 2019; in final form April 2, 2020; Published online April 18, 2020
Citation:
Yuma Mizuno, “Exponents Associated with $Y$-Systems and their Relationship with $q$-Series”, SIGMA, 16 (2020), 028, 42 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1565 https://www.mathnet.ru/eng/sigma/v16/p28
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Abstract page: | 111 | Full-text PDF : | 38 | References: | 31 |
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