Note on Exponents Associated with Y-Systems

TL;DR

This paper proves Mizuno's conjecture for exponents in level 2 Y-systems of classical Dynkin types B and D, and reformulates it for type C, advancing understanding of algebraic relations in these systems.

Contribution

The paper provides proofs for Mizuno's conjecture on exponents in level 2 Y-systems for types B and D, and offers a new formulation for type C.

Findings

Proof of Mizuno's conjecture for (B_n,2)

Proof of Mizuno's conjecture for (D_n,2)

Reformulation of the conjecture for (C_n,2)

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 $(C_n,2)$.