Title: Permutation mixing via hypercontractivity for symmetric characters
Abstract:
The area of permutation mixing asks, for a given set of permutations B \subseteq S_n, whether the product x_1 \dots x_k is approximately uniformly distributed, where all x_i are uniformly sampled from B. Permutation mixing has been extensively studied over the past few decades, starting with the seminal work of Diaconis and Shahshahani in 1981 on deck shuffling. One major problem in this area asks to determine the minimal size of a conjugacy-closed set B that ensures it has a mixing-time of k, i.e., that multiplying k uniform elements of B results in mixing. Larsen and Shalev (Inventiones Math., 2008) found an almost tight bound on this size, employing novel methods to bound the characters of the symmetric group. These bounds rely heavily on specific properties of the symmetric group and its character formulas.
In this talk, we present a new approach to bounding symmetric characters, drawing on a tool from the theory of Boolean functions known as 'hypercontractivity for global functions'. This method improves the bounds obtained by Larsen-Shalev, and relies on only a few key properties of symmetric characters --- properties that have analogues in various classical groups. This raises the possibility of extending the technique to other groups. After introducing the relevant tools from representation theory, we will outline the proof of the new bound and deduce improved bounds on the size that ensures a set has mixing time k.
The talk is based on joint work with Noam Lifshitz. No prior knowledge is required.