Analysis Seminar

Wednesday, April 6, 2022 4:00 pm - 4:00 pm EDT

Jacob Campbell, Department of Pure Mathematics, University of Waterloo

"Characters of the infinite symmetric group and random matrices"

A famous theorem of Thoma from 1964 parameterizes the extremal characters of the infinite symmetric group $S_{\infty}$ in terms of certain scalar sequences. An extremal character of $S_{\infty}$ gives a good example of a trace on its group algebra, and allows one to consider permutations as noncommutative random variables. On the other hand, in the last few decades the so-called Jucys-Murphy elements of the group algebra, which are related to the star-transpositions $(1,n)$, have played a central role in the representation theory of symmetric groups.

In this talk, I will present joint work with C. Koestler and A. Nica (arXiv:2203.01763) where we prove a central limit-type theorem for the star-transpositions, with respect to the trace coming from certain Thoma sequences. For the Thoma sequence (0,0,...), i.e. in the regular representation, the limit is semicircular, as already found by Biane in 1995. More recently, this was generalized by Koestler-Nica in 2021 for a sequence (1/d,...,1/d,0,0,...); in this case, the limit law is the average eigenvalue distribution of a certain well-known random matrix called the d x d traceless GUE.

We extend the latter result to look at all finite Thoma sequences $(w_1,...,w_d,0,0,...)$ and land on something unexpected: the limit law comes from a GUE-like random matrix whose off-diagonal entries live in noncommutative algebras with canonical commutation relations (CCR). I will explain how Thoma characters and star-transpositions naturally point towards the combinatorics of GUEs, through the probabilistic notion of exchangeability and the so-called "genus expansion" for the GUE, and how the off-diagonal noncommutativity is introduced by varying the Thoma parameter.

This seminar will be held jointly online and in person: