Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 028, 42 pages      arXiv:1812.05863      https://doi.org/10.3842/SIGMA.2020.028
Contribution to the Special Issue on Cluster Algebras

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

Received September 27, 2019, in final form April 02, 2020; Published online April 18, 2020

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.

Key words: cluster algebras; $Y$-systems; root systems; $q$-series.

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