Stability of the centers of group algebras of $G{L}_n(q)$

Abstract

The center ${\mathcal{Z}}_n(q)$ of the integral group algebra of the general linear group $G{L}_n(q)$ over a finite field admits a filtration with respect to the reflection length. We show that the structure constants of the associated graded algebras ${\mathcal{G}}_n(q)$ are independent of $n$, and this stability leads to a universal stable center with positive integer structure constants which governs the algebras ${\mathcal{G}}_n(q)$ for all $n$. Various structure constants of the stable center are computed and several conjectures are formulated. Analogous stability properties for symmetric groups and wreath products were established earlier by Farahat-Higman and the second author.

V3; 27 pages; a remark updated

References in corpus (3)

• Absolute order in general linear groups
• Quantum integer-valued polynomials
• Anti-commuting varieties

Cited by in corpus (4)

• Anti-commuting varieties
• Stability of the Hecke algebra of wreath products
• Stable Centres II: Finite Classical Groups
• Stability of the centers of the symplectic groups rings $\matbb{Z}[Sp_{2n}(q)]$