The Anatomy of Induced Modular Representations via Brauer Characters

Modular representation theory is a vibrant field in algebra that studies the actions of finite groups on vector spaces over fields whose characteristic divides the order of the group. Unlike ordinary representation theory, modular representations are not necessarily semisimple, leading to a richer and more complex structure. Induced representations are a fundamental construction, allowing the ascent of representations from subgroups to the full group, and their properties are crucial for understanding the global representation theory of a group. Brauer characters provide a powerful bridge between modular representations and the more tractable ordinary characters, allowing for a character-theoretic approach to phenomena in positive characteristic. This paper delves into the anatomy of induced modular representations, exploring their structure and properties as revealed through the lens of Brauer characters. We will systematically define modular representations, their induction, and Brauer characters, investigating how the Brauer character of an induced modular representation can be computed and what insights it offers into the module's decomposition and structure. Special attention will be paid to the Mackey formula in the modular context and its implications for understanding the direct summands of induced modules. This study offers a unique synthesis, providing a clear and unified character-theoretic framework to diagnose and understand the intricate internal structure of induced modular representations, thereby clarifying their roles in block theory and the classification of finite groups. The interplay between these concepts provides a deep understanding of how local information from subgroups contributes to the overall representation theory of the finite group, shedding light on the intricate connections between characteristic $p$ and characteristic 0 theories.