Motivated by the study of simultaneous cores, we give three proofs (in varying levels of generality) that the expected norm of a weight in a highest weight representation $V_{\lambda }$ of a complex simple Lie algebra $\mathfrak{g}$ is $\frac{1}{h+1}〈\lambda ,\lambda +2\rho 〉$. First, we argue directly using the polynomial method and the Weyl character formula. Second, we relate this problem to the “Winnie-the-Pooh problem” regarding orthogonal decompositions of Lie algebras; although this approach offers the most explanatory power by interpreting the quantity $〈\lambda ,\lambda +2\rho 〉$ as the eigenvalue for the Casimir element on $V_{\lambda }$, it applies only to Cartan types other than A and C. Third, we use the combinatorics of semistandard tableaux to obtain the result in type A. We conclude with computations of many combinatorial cumulants.

通过同时核的研究，我们给出了三个证明（在不同的普遍性水平下），即最大重量表示形式下的预期重量范数 $V_{\lambda }$ 复杂的简单李代数 $\mathfrak{G}$ 是 $\frac{\mathrm{1个}}{H+\mathrm{1个}}〈\lambda ，\lambda +2\rho 〉$。首先，我们直接使用多项式方法和Weyl字符公式进行争论。其次，我们将此问题与有关李代数正交分解的“维尼小熊维尼问题”联系起来。尽管这种方法通过解释数量提供了最具解释力$〈\lambda ，\lambda +2\rho 〉$ 作为Casimir元素的特征值 $V_{\lambda }$，它仅适用于A和C以外的Cartan类型。第三，我们使用半标准表格的组合来获得类型A的结果。我们以许多组合累积量的计算得出结论。