On the lifting of the Dade group

TL;DR

This paper proves that for the Dade group of endo-permutation modules over a finite p-group, the reduction map from a characteristic 0 ring to its residue field admits a group homomorphism section, ensuring a structural lift.

Contribution

It establishes the existence of a group homomorphism section for the reduction map in the Dade group, a novel result in modular representation theory.

Findings

The reduction map always admits a group homomorphism section.

This result applies to endo-permutation modules over finite p-groups.

It enhances understanding of the lifting problem in modular representation theory.

Abstract

For the group of endo-permutation modules of a finite \(p\)-group, there is a surjective reduction modulo \(p\) homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic \(p\). We prove that this reduction map always has a section which is a group homomorphism.