Exponents Associated with Y-Systems and their Relationship with q-Series

Abstract

Let 𝘟ᵣ be a finite type Dynkin diagram, and ℓ be a positive integer greater than or equal to two. The 𝘠-system of type 𝘟ᵣ with level ℓ is a system of algebraic relations whose solutions have been proved to have periodicity. For any pair (𝘟ᵣ,ℓ), we define an integer sequence called exponents using the formulation of the 𝘠-system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type 𝘟ᵣ, and prove this conjecture for (A₁,ℓ) and (Aᵣ,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.