Farahat-Higman Algebras for Classical Linear Groups

Abstract:

In 1959, Farahat and Higman showed a remarkable property about the symmetric group: the structure constants for the center of the group algebra $S_n$ depend polynomially on $n$ and that there exists an infinite-dimensional algebra, which we call the Farahat-Higman algebra, that sits over these centers. Moreover, these admit mutually compatible filtrations such that the structure constants of the associated graded rings do not depend on $n$ at all.

In 2003, Ivanov and Kerov came up with a new and elegant way to derive the results of Farahat and Higman. In this talk we review all of these results and then show how we can take an approach similar to that of Ivanov and Kerov to prove statements analogous to those above but for the classical linear groups over finite fields, like $G{L}_n(F_q)$. Time permitting we will also discuss some connections to a notion called a symmetric tensor category, which can be thought of as a generalization of a representation category of a group.

Based on joint work (https://arxiv.org/pdf/2112.01467.pdf) with Christopher Ryba.

Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.

Join Zoom Meeting: https://washington.zoom.us/j/91547335974
Meeting ID: 915 4733 5974