Asymptotics of Characters of Symmetric Groups and Free Probability

In order to answer the question “what is the asymptotic theory of representations of Sn” we will present two concrete problems. In both cases the solution requires a good understanding of the product (convolution) of conjugacy classes in the symmetric group and we will present a combinatorial setup for explicit calculation of such products. The asymptotic behavior of each summand in our expansion will depend on topology (genus) of a two-dimensional surface associated to some partitions and for this reason our method carries a strong resemblance to the genus expansion from the random matrix theory. In particular, non-crossing partitions and free probability play a special role. 1. WHAT IS THE ASYMPTOTIC THEORY OF THE REPRESENTATIONS OF THE SYMMETRIC GROUPS Sn? 1.1. (Generalized) Young diagrams. Irreducible representations ρ of the symmetric group Sn are in a one-to-one correspondence with Young diagrams λ having n boxes. An example of a Young diagram is presented on Figure 1.1. This figure also explains the notion of a profile of a Young diagram. For a Young diagram with n boxes the area of the shaded region is equal to 2n. After we shrink the geometric representation of this Young diagram by by factor 1 √ n we obtain a generalized Young diagram (cf Figure 1.2) for which the area of the shaded region is equal to 2. In the following we will compare the shapes of the Young diagrams only after such a rescaling. Please note that the usual definition of a Young diagram λ says that it is a weakly decreasing sequence λ1 ≥ λ2 ≥ · · · ≥ λk of positive integers and the generalized Young diagram considered above do not fit into this category. Instead, any generalized Young diagram is by definition identified with its profile (i.e. a function f : R → R+ with some additional constraints which an interested Reader can easily guess) [Ker93]. 1.2. Example of a problem: characters of a large Young diagram. Let (λn) be a sequence of Young diagrams such that λn has n boxes and that the shapes of the Young diagrams λn converge (after rescaling) to some generalized Young diagram.