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 $D^{\Omega }(G)$ of $D(G)$ generated by relative syzygies ${\Omega }_X$, 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 ${\omega }_X$ of $C(G,p)$, and we prove the existence of a well-defined group homomorphism ${\Psi }_G:C(G,p)\to D^{\Omega }(G)$ that sends ${\omega }_X$ to ${\Omega }_X$. 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 ${\Psi }_G$.

Minor revision, 40 pages