Strange Expectations and the Winnie-the-Pooh Problem

Motivated by the study of simultaneous cores, we give three proofs (in varying levels of generality) for the expected norm of a weight in a highest weight representation of a complex simple Lie algebra. First, we argue directly using the polynomial method and the Weyl character formula. Second, we use the combinatorics of semistandard tableaux to obtain the result in type A. Third, and most interestingly, we relate this problem to the "Winnie-the-Pooh problem" regarding orthogonal decompositions of Lie algebras; although this approach offers the most explanatory power, it applies only to Cartan types other than A and C. We conclude with computations of many combinatorial cumulants.