On the lifting problem of representations of a metacyclic group

TL;DR

This paper establishes a precise criterion for lifting modular representations of certain metacyclic groups from characteristic zero fields to local principal ideal domains, enhancing understanding of their algebraic structure.

Contribution

It provides a necessary and sufficient condition for lifting representations of metacyclic groups, a novel result in the theory of modular representations.

Findings

Derived a criterion for lifting representations of metacyclic groups

Connected representation lifting to roots of unity in local domains

Clarified structural conditions for modular representation liftings

Abstract

We give a necessary and sufficient condition for a modular representation of a group $G=C_{p^h} \rtimes C_m$ in a field of characteristic zero to be lifted to a representation over local principal ideal domain of characteristic zero containing the $p^h$ roots of unity.