Modular Representations, Old and New

TL;DR

This survey reviews the historical and recent advances in modular representation theory of finite groups, highlighting conjectures, developments in reductive groups, and connections with Lie theory and graded representations.

Contribution

It provides a comprehensive overview of both classical and recent progress, emphasizing new links with Lie theory and graded representation theory.

Findings

Progress on outstanding conjectures in modular representation theory

Development of modular representations in non-defining characteristic of finite reductive groups

Recent connections between symmetric groups, Hecke algebras, and Lie theory

Abstract

The modular representation theory of finite groups has its origins in the work of Richard Brauer. In this survey article we first discuss the work being done on some outstanding conjectures in the theory. We then describe work done in the eighties and nineties on modular representations in non-defining characteristic of finite reductive groups. In the second part of the paper we discuss some recent developments in the theory for symmetric groups and Hecke algebras, where remarkable connections with Lie theory and graded representation theory have been made.