Gaussian Fluctuations of Representations of Wreath Products

We study the asymptotics of the reducible representations of the wreath productsG≀Sq = G⋊Sq for large q, whereG is a fixed finite group and Sq is the symmetric group in q elements; in particular for G = Z/2Z we recover the hyperoctahedral groups. We decompose such a reducible representation of G ≀ Sq as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov’s central limit theorem for wreath products.