# The largest coefficient of the highest root and the second smallest exponent

## TL;DR

This paper explores the relationship between the second smallest exponent of Weyl groups and the largest coefficient of the highest root, providing a uniform proof and a characterization of the G2 root system.

## Contribution

It offers a uniform, elementary proof connecting the second smallest exponent with the highest root coefficient, and characterizes G2 root systems using these numbers.

## Key findings

- The second smallest exponent is one or two plus the largest coefficient of the highest root.
- A necessary and sufficient condition for G2 root systems is established.
- The proof simplifies understanding of exponents and root coefficients in Weyl groups.

## Abstract

There are many different ways that the exponents of Weyl groups of irreducible root systems have been defined and put into practice. One of the most classical and algebraic definitions of the exponents is related to the eigenvalues of Coxeter elements. While the coefficients of the height root when expressed as a linear combination of simple roots are combinatorial objects in nature, there are several results asserting relations between these exponents and coefficients. This study was conducted to give a uniform and fairly elementary proof of the fact that the second smallest exponent of the Weyl group is one or two plus the largest coefficient of the highest root of the root system depending upon a simple condition on the root lengths. As a consequence, we obtain a necessary and sufficient condition for a root system to be of type $G_2$ in terms of these numbers.