Dade groups for finite groups and dimension functions

Abstract

Let G be a finite group and k an algebraically closed field of characteristic p > 0. We define the notion of a Dade kG-module as a generalization of endo-permutation modules for p-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade kG-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group D(G) defined by Lassueur. We also consider the subgroup DΩ(G) of D(G) generated by relative syzygies ΩX , where X is a finite G-set. If C(G, p) denotes the group of superclass functions defined on the p-subgroups of G, there are natural generators ωX of C(G, p), and we prove the existence of a well-defined group homomorphism ΨG : C(G, p) → DΩ(G) that sends ωX to ΩX . The main theorem of the paper is the verification that the subgroup of C(G, p) consisting of the dimension functions of k-orientable real representations of G lies in the kernel of ΨG.