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

Let R = Z 2 2 f ?1 ] and G = SL 2 (2 f). The group ring RG is calculated nearly up to Morita equivalence. In particular the irreducible RG-lattices can be described purely combinatorically in terms of subsets of f1 The group G = SL 2 (p f) of all 22-matrices over the eld k with p f elements is one of the simplest examples of a nonabelian nite group of Lie type. Its representation theory in characteristic 0 was already investigated by I. Schur 13] and its modular representation theory is also well understood ((1], 2]). The next step is to describe the integral group ring RG of G when R is the ring of integers in a nite extension of the eld Q l of l-adic numbers, to bring together the characteristic 0 and the characteristic l information. If l 6 = p and l 6 = 2 then the defect groups of the ring direct summands of RG are cyclic, so RG is described by the general theory of blocks with cyclic defect groups ((10], 12], 7]). For odd primes p the Sylow 2-subgroups of G are dihedral groups and 10], Chapter VII investigates RG for l = 2. So the only remaining case is l = p, where the Sylow p-subgroups of G are elementary abelian of rank f. If f = 1 one again has the cyclic defect case and for f = 2 the group ring Z p G is described up to Morita equivalence in 8]. In the present paper the remaining cases f 3 are treated for p = 2. To nd kG, one uses methods from the representation theory for groups of Lie type in deening characteristic. However these methods are not directly applicable for calculating RG, when R = Z 2 2 f ?1 ] is the ring of integers in the unramiied extension K of degree f of Q 2. The new idea used in this paper is to start from the explicit presentation of kG given in 6] and lift the generators of kG to generators of RG. The explicit knowledge of kG together with the decomposition numbers calculated in 4] and 5] do not seem to be suucient to determine RG up to Morita equivalence. But they give enough information to describe the inclusion patterns of the irreducible RG-lattices (Theorem 3.15) as well as …