Strange expectations and the Winnie-the-Pooh problem

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 g is 1/h9+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. (C) 2020 Elsevier Inc. All rights reserved.