Work

Note on Exponents Associated with Y-Systems

Year: 2025

Type: article

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 $(C_n,2)$.

Source: Symmetry Integrability and Geometry Methods and Applications

DOI: 10.3842/sigma.2025.028

Author: Ryo Takenaka

Institution: Osaka Metropolitan University

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Ministry of Education, Culture, Sports, Science and Technology

Award: G1106070903