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

## TL;DR

This paper explores the exponents associated with $Y$-systems of finite type Dynkin diagrams, proposing a conjectural formula linked to root systems and $q$-series, with proofs for specific cases and implications for affine Lie algebra modules.

## Contribution

It introduces a conjectural formula relating $Y$-system exponents to root systems, proves it for certain cases, and connects these exponents to $q$-series identities and affine Lie algebra modules.

## Key findings

- Conjectural formula for exponents in $Y$-systems based on root systems.
- Proof of the conjecture for $(A_1, \, \ell)$ and $(A_r, 2)$ cases.
- Relationship established between exponents and asymptotic dimensions of affine Lie algebra modules.

## Abstract

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