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 endopermutation 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 $$.
We also consider the subgroup $D^\Omega (G)$ of $D(G)$ generated by relative syzygies $\Omega X$, where $X$ is a finite $G$-set. Let $C(G; p)$ denote the group of superclass functions defined on the p-subgroups of G. There are natural generators $\omega_X$ of $C(G; p)$. We prove that there is a well-defined group homomorphism $\psi_G : C(G; p) \to D^\Omega (G)$ that sends $\omega_X$ to $ \Omega_X$.
The main theorem is the verification that the subgroup of $C(G; p)$ consisting of the dimension functions of $k$-orientable real representations of $G$ lies in the kernel of $\psi_G$. In the proof we consider Moore $G$-spaces which are the equivariant versions of spaces which have nonzero reduced homology in only one dimension.
This talk is about a theorem in modular representation theory whose proof is topological using equivariant homotopy theory and homological algebra over orbit category. I will give all necessary definitions to make it possible to follow the talk and provide examples to motivate the theorems. This is a joint work with Matthew Gelvin $$.
$$ M. Gelvin and E. Yalçın, Dade Groups for Finite Groups and Dimension Functions, preprint, 2020 (arXiv:2007.05322v2).
$$ C. Lassueur, The Dade group of a finite group, J. Pure Appl. Algebra, 217 (2013), 97-113.