Stable Centres II: Finite Classical Groups

TL;DR

This paper extends the concept of polynomial structure constants from symmetric groups to classical groups over finite fields, constructing analogous algebras for groups like GL, U, Sp, and O.

Contribution

It introduces a new algebraic framework for classical groups over finite fields, generalizing the Farahat-Higman algebra concept beyond symmetric groups.

Findings

Structure constants are polynomial in q^n for classical groups.

Constructed algebraic analogs of Farahat-Higman algebra for these groups.

Provides a unified approach to centers of group algebras across different groups.

Abstract

Farahat and Higman constructed an algebra $\mathrm{FH}$ interpolating the centres of symmetric group algebras $Z(\mathbb{Z}S_n)$ by proving that the structure constants in these rings are "polynomial in $n$". Inspired by a construction of $\mathrm{FH}$ due to Ivanov and Kerov, we prove for $G_n = GL_n, U_n, Sp_{2n}, O_n$, that the structure constants of $Z(\mathbb{Z}G_n(\mathbb{F}_q))$ are "polynomial in $q^n$", allowing us to construct an equivalent of the Farahat-Higman algebra in each case.