We investigate the structure constants of the center ${\mathcal{H}}_n$ of the group algebra $\mathbb{Z}[S{p}_n(q)]$ over a finite field. The reflection length on the group $G{L}_{2n}(q)$ induces a filtration on the algebras ${\mathcal{H}}_n$. We prove that the structure constants of the associated graded algebra ${\mathcal{S}}_n$ are independent of n. As tool in the proof we consider the embedding $S{p}_m\hookrightarrow S{p}_n(q)$ and determine the behavior of the centralizers and intersection of centralizers under the embedding $S{p}_m\hookrightarrow S{p}_n(q)$. We determine explicit formulas for rate of the growth of the centralizer of an element $U\in S{p}_m(q)$ in terms of dimension of the fixed space of U.

中文翻译：

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辛群环中心的稳定性 $\mathbb{ž}[\mathrm{小号}{p}_{2ñ}（q）]$

我们研究中心的结构常数 ${\mathcal{H}}_{ñ}$ 群代数 $\mathbb{ž}[\mathrm{小号}{p}_{ñ}（q）]$在一个有限的领域。在组上的反射长度$G{\mathrm{大号}}_{2ñ}（q）$ 诱导代数上的过滤 ${\mathcal{H}}_{ñ}$。我们证明了相关梯度代数的结构常数${\mathcal{小号}}_{ñ}$独立于n。作为证明的工具，我们考虑嵌入$\mathrm{小号}{p}_{米}\hookrightarrow \mathrm{小号}{p}_{ñ}（q）$ 并确定嵌入下扶正器和扶正器交点的行为 $\mathrm{小号}{p}_{米}\hookrightarrow \mathrm{小号}{p}_{ñ}（q）$。我们确定元素扶正剂增长率的明确公式$ü\in \mathrm{小号}{p}_{米}（q）$根据U的固定空间的尺寸。