Dade Groups for Finite Groups and Dimension Functions

TL;DR

This paper introduces Dade $kG$-modules as a generalization of endo-permutation modules for finite groups, establishing their group structure, and explores connections with dimension functions of real representations.

Contribution

It defines Dade $kG$-modules for arbitrary finite groups, constructs a related group isomorphism, and links dimension functions of real representations to the Dade group via a homomorphism.

Findings

Dade $kG$-modules form a group under tensor product.

A homomorphism from superclass functions to the Dade group is constructed.

Dimension functions of $k$-orientable real representations lie in the kernel of this homomorphism.

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 $\Psi_G$.