The Universal Property of Induced Brauer Characters

This paper rigorously investigates the universal property governing induced Brauer characters in the context of modular representation theory of finite groups. Brauer characters serve as indispensable tools for analyzing representations over fields of positive characteristic, where the classical theory of ordinary characters often falters due to the non-semisimplicity of group algebras. While the concept of induced characters and their universal properties are well-established in characteristic zero via Frobenius reciprocity, their modular counterparts require careful reformulation. We establish a formal definition for induced Brauer characters and subsequently prove a universal property that characterizes them uniquely up to isomorphism. This property, expressed through a natural isomorphism of Hom-spaces, demonstrates a fundamental adjointness relationship between induction and restriction functors at the level of modules, which then translates to the character theory. Our methodology involves constructing induced modules via tensor products and then deriving their Brauer characters, demonstrating how these constructions align with the stated universal property. The findings deepen our understanding of the structural behavior of Brauer characters under induction, offering crucial insights into the intricate landscape of modular representation theory and providing a solid foundation for further exploration into phenomena like block theory and decomposition matrices.