Stability of the centers of group algebras of GL(q)

Abstract

The center Zn(q) of the integral group algebra of the general linear group GLn(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 Gn(q) are independent of n, and this stability leads to a universal stable center with positive integer structure constants which governs the algebras Gn(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.2000 Mathematics Subject Classification. Primary: 20G40, 05E15.

Stable Centres II: Finite Classical Groups

“…This follows from Lemma 8.6 (see the proof of Proposition 6.10). This addresses one of the further directions recommended in [WW19], and also in [Mé10]. 8.2.…”

“…Proposition 6.10 (Theorem 3.4, [WW19]). The algebra FH GL q is filtered, where K µ is in filtration degree |µ|.…”

“…First of all, we need a way to discuss conjugacy classes of these groups for all n at once. Definition 6.1 (Subsection 2.3 [WW19]). If g ∈ GL n (F q ), we define the modified type of g to be the multipartition ν obtained from the type, µ, of g in the following way.…”

“…In this paper we construct an analogous version of FH for classical groups G n , interpolating the centres Z(ZG n (F q )). For the case G n = GL n , an analogue of reduced cycle type for the general linear group called modified type was defined by Wan and Wang [WW19]. If X µ is the sum of all elements of GL n (F q ) of modified type µ, let the structure constants a λ µν (n) be defined by…”

“…Issues have been raised with the proof which we do not know how to address; see Remark 3.6 of[WW19].…”