Strange expectations and the Winnie-the-Pooh problem

Abstract

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 Vλ of a complex simple Lie algebra g is 1h+1〈λ,λ+2ρ〉. 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 〈λ,λ+2ρ〉 as the eigenvalue for the Casimir element on Vλ, 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.