Brauer's generalized decomposition numbers and universal deformation rings

TL;DR

This paper introduces a new technique using Brauer's generalized decomposition numbers to determine the universal deformation rings of mod p representations of finite groups, enhancing understanding of their liftings.

Contribution

The paper presents a novel method leveraging Brauer's generalized decomposition numbers to explicitly compute universal deformation rings for finite group representations.

Findings

New technique for computing R(G,V) using Brauer's numbers

Explicit descriptions of deformation rings for certain finite groups

Improved understanding of lifting properties of mod p representations

Abstract

The versal deformation ring R(G,V) of a mod p representation V of a profinite group G encodes all isomorphism classes of lifts of V to representations of G over complete local commutative Noetherian rings. We introduce a new technique for determining R(G,V) when G is finite which involves Brauer's generalized decomposition numbers.