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 highest 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 bijective 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.

定义和实施不可还原根系的Weyl基的指数有许多不同的方法。指数的最经典和代数定义之一与Coxeter元素的特征值有关。当最高根的系数表示为简单根的线性组合时，实际上是组合对象，但有一些结果证明了这些指数与系数之间的关系。进行该研究以给出一致且双射的证据，这一事实表明，取决于根部长度的简单条件，Weyl基的第二最小指数是一或二加上根系最高根的最大系数。结果，我们获得了根系统为类型的必要和充分条件\（G_2 \）就这些数字而言。