The Group Ring of SL2(pf) over p-adic Integers for p Odd

Abstract

This paper describes the ring-theoretic structure of the group rings of SL p 2 over the p-adic integers.

The p-adic group ring of SL_2(p^f)

“…This order has (by construction) semisimple K-span, the same decomposition matrix as the basic order of O SL 2 (2 f ) and it is self-dual with respect to the appropriate trace bilinear form. Moreover, it reduces to an k-algebra which, upon tensoring with k, becomes isomorphic to the basic algebra of k SL 2 (2 f ) as described by Koshita. In the present article we confirm this conjecture, as well as the analogous conjecture concerning the group ring of SL 2 (p f ) proposed in [Neb00b].…”

“…This article is concerned with the group ring O SL 2 (p f ) for some f ∈ N. Hence we are dealing with the discrete valuation ring version of what is typically referred to as representation theory in "defining characteristic". Our aim in this paper is to prove a conjecture made by Nebe in [Neb00a] (for the case p = 2) respectively [Neb00b] (for the case p odd) which claims (rightly) to describe the group ring of SL 2 (p f ) over sufficiently large extensions O of Z p . Here, "to describe the group ring" means to describe its basic order.…”

On Group Ring Automorphisms

“…We verify this criterion for the perfect group of order 1080, which is a central extension of the alternating group A 6 by a cyclic group of order 3, in Section 5. This will follow from general considerations as soon as we have established that Outcent has been given in [13]. Furthermore, Outcent ¥ ¤ is, with regard to p-adic group rings, invariant under Morita equivalence.…”

“…For a finite group G, the group Outcent £ ¥ ¤ p G¦ of outer central automorphisms of ¤ p G only depends on the Morita equivalence class of ¤ p G, which allows reduction to a basic order for its calculation. If the group ring is strongly related to a graduated order, it is often possible to give an explicit description of the basic order (see [14,13] …”

The group ring of SL2(2f) over 2-adic integers

“…There are additional technical di½culties for odd primes p that make the basic ideas less transparent, so the case p b 2 is treated in a separate paper [10]. Let V be a simple KG-module of dimension n and let M I 1 Y F F F Y M I r be the 2-modular constituents of V of dimension n j X 2 jI j j dim k M I j , 1 j r. Then there is a basis of V such that the corresponding matrix representation h V satis®es h V RG fX ij 1iY jr e R nÂn j X ij e 2 jI i ÀI j j R n i Ân j gX This is the ®rst time that such a detailed description of an in®nite series of p-adic group rings has been found, where not only the order but also the number of generators of the Sylow p-subgroups grows.…”