Induction and Structure Theorems for Grothendieck and Witt Rings of Orthogonal Representations of Finite Groups

The Grothendieckand Wittring of orthogonal representations of a finite group is defined and studied. The main application (only indicated) is the reduction of the computation of Wall's various L-groups for a finite group n to those subgroups of n, which are a semidirect product of a cyclic group y of odd order with a 2-group /?, such that any element in ft acts on y either by the identity or by taking any element in to its inverse. Let 7i be a finite group and R a Dedekind ring. An Krc-lattice (M, ƒ) or just M is defined to be a finitely generated, K-projective jR7i-module M together with a symmetric, 71-invariant nonsingular form ƒ : M x M -> R (cf. [3]). For two ^71-lattices Mx and M2 one has their orthogonal sum M1 J_ M2 and tensor product Mx ® M2, thus the isomorphism classes of Rrc-lattices form a half-ring Y(R, n\ whose associated Grothendieck ring is denoted by Y(R, n). For a subgroup y ^ n one has in an obvious way, restriction and induction of JR7i-lattices, resp. jRy-lattices, and it is easily seen (cf. [3]) that this makes Y(Rr) into a G-functor in the sense of Green (cf. [5]). As in the theory of integral group-representations, where the Grothendieck ring of isomorphism classes of i^7i-modules is much too large for many purposes and is thus replaced by its quotient G0{R, n) (in the sense of [9]) modulo the ideal, generated by the Euler characteristics of short exact sequences of ita-modules, we are going to define certain quotients of Y(R,n), using a relation which was first introduced by D. Quillen in [7, §5]. At first let us remark, that for any finitely generated R~ projective /^-module N, one has the associated hyperbolic module H(N) = (AT 0 N*9 ƒ ) with AT* = HomK(iV, R) the K-dua! of AT, considered as K7c-module (with (g • v)(n) = v{g~ • n\ g e TT, V e N*9 n e iV).and f(N9N) = /(N*,N*) = 0, f(n,v) = v(n\ neN, veN*. We now define a Quillen pair (M, N) to be an Kyr-lattice M = (M, ƒ ) together with an AM S (MOS) subject classifications (1970). Primary 15A63, 16A54, 18F25; Secondary 20C99, 20J99, 57D65.