Quadratic Geometries, Projective Modules, and Idempotents Original Research Article

Let G be a finite group and let k be a field. We say that a kG-module V has a quadratic geometry or is of quadratic type if there exists a non-degenerate (equivalently non-singular) G-invariant quadratic form on V. If V is irreducible or projective indecomposable and k of odd characteristic, the existence of a quadratic geometry can be read off from invariants of representation theory. In terms of the group algebra the existence of such a geometry is characterized by the existence of an idempotent e which corresponds to V and is invariant under the antiautomorphism on kG given by g → = g−1 for g set membership, variant G. In characteristic two the problem is more delicate. For an irreducible module the existence of a quadratic geometry is partly a question in cohomology. Unfortunately, a satisfactory answer does not seem to be known. However, if V is projective, liftability of the module to characteristic zero may help to find the geometry more easily. This is the subject of the paper. The notation which we shall use throughout the paper is standard and may be taken from Huppert and Blackburn ("Finite Groups II," Springer-Verlag, Berlin, 1982) and Isaacs ("Character Theory of Finite Groups," Academic Press, New York, 1976).