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

Let $X_r$ be a finite type Dynkin diagram, and $\mathrm{\ell }$ be a positive integer greater than or equal to two. The $Y$-system of type $X_r$ with level $\mathrm{\ell }$ is a system of algebraic relations, whose solutions have been proved to have periodicity. For any pair $(X_r,\mathrm{\ell })$, 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 $(A_1,\mathrm{\ell })$ 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.