Stability of the centers of group algebras of general affine groups GA_n(q)

2025/06/06 by Wan, Jinkui, Zhou, Lan
#05E15 #20G40 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2506.05652

Abstract

The general affine group GA_n(q) consisting of invertible affine transformations of an affine space of codimension one in the vector space \mathbbF_qⁿ over a finite field \mathbbF_q, can be viewed as a subgroup of the general linear group GL_n(q) over \mathbbF_q. In the article, we introduce the notion of the type of each matrix in GA_n(q) and give an explicit representative for each conjugacy class. Then the center \mathscrA_n(q) of the integral group algebra ℤ[GA_n(q)] is proved to be a filtered algebra via the length function defined via the reflections lying in GA_n(q). We show in the associated graded algebras \mathscrG_n(q) the structure constants with respect to the basis consisting of the conjugacy class sums are independent of n. The structure constants in \mathscrG_n(q) is further shown to contain the structure constants in the graded algebras introduced by the first author and Wang for GL_n(q) as special cases. The stability leads to a universal stable center \mathscrG(q) with positive integer structure constants only depending on q which governs the algebras \mathscrG_n(q) for all n.

Citations

• Stable Centres II: Finite Classical Groups
• Ideals of the cohomology rings of Hilbert schemes and their applications
• The Farahat-Higman ring of wreath products and Hilbert schemes
• Symmetric groups and the cup product on the cohomology of Hilbert schemes
• Conjugacy classes in maximal parabolic subgroups of general linear groups

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