# Stability of the centers of the symplectic groups rings $\matbb{Z}[Sp_{2n}(q)]$

## TL;DR

This paper studies the structure constants of the center of the group algebra of symplectic groups over finite fields, showing their independence from the group size and analyzing centralizer growth under embeddings.

## Contribution

It proves the independence of structure constants from group size for the filtered algebra associated with symplectic groups and determines the growth of centralizers under embeddings.

## Key findings

- Structure constants are independent of n.
- Centralizer index formula under embeddings.
- Growth of centralizers in symplectic groups.

## Abstract

We investigate the structure constants of the center $\mathcal{H}_n$ of the group algebra $Sp_{n}(q)$ over a finite field. The reflection length on the group $GL_{2n}(q)$ induces a filtration on the algebras $\mathcal{H}_n$. We prove that the structure constants of the associated filtered algebra $\mathcal{S}_n$ are independent of $n$. As a technical tool in the proof, we determine the growth of the centralizers under the embedding $Sp_m(q)\subset Sp_{m+l}(q)$ and we show that the index of the centralizer of $g\in Sp_m(q)$ in the centralizer of $g\in Sp_{m+k}$ is equal to $q^{2ld}|Sp_{r+l}(q)||Sp_{r}(q)|^{-1}$ for some $d$ and $r$ which are uniquely determined by the conjugacy class of $g$ in $GL_{2n}(q).$