Kerov’s central limit theorem for Schur-Weyl measures of parameter 1/2

Abstract. In this article, we show that Kerov’s central limit theorem related to the fluctuations of Young diagrams under the Plancherel measure extends to the case of Schur-Weyl measures, which are the probability measures on partitions associated to the representations of the symmetric groups Sn on tensor products of vector spaces V ⊗n (see [Bia01]). The proof is inspired by the one given in [IO02] for the Plancherel measure, and it relies on the combinatorics of the algebra of observables of diagrams; we shall also use Śniady’s theory of cumulants of observables (cf. [Ś06]). Given a finite groupG and a finite-dimensional complex linear representation V ofG, the decomposition in irreducible components V = λ∈Ĝmλ Vλ yields a probability measure on the set G ̂ of isomorphism classes of irreducible representations of G