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

SIGMA 21 (2025), 028, 21 pages      arXiv:2410.02286      https://doi.org/10.3842/SIGMA.2025.028

Note on Exponents Associated with Y-Systems

Ryo Takenaka
Department of Mathematics, Osaka Metropolitan University, Osaka 558-8585, Japan

Received October 14, 2024, in final form April 16, 2025; Published online April 24, 2025

Abstract
Let $(X_n,\ell)$ be the pair consisting of the Dynkin diagram of finite type $X_n$ and a positive integer $\ell\geq2$, called the level. Then we obtain the Y-system, which is the set of algebraic relations associated with this pair. Related to the Y-system, a sequence of integers called exponents is defined through a quiver derived from the pair $(X_n,\ell)$. Mizuno provided conjectured formulas for the exponents associated with Y-systems in [Mizuno Y., SIGMA 16 (2020), 028, 42 pages, arXiv:1812.05863]. In this paper, we study the exponents associated with level 2 Y-systems for classical Dynkin types. As a result, we present proofs of Mizuno's conjecture for $(B_n,2)$ and $(D_n,2)$, and give a reformulation for $(C_n,2)$.

Key words: cluster algebras; Y-systems; root systems.

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