The Clifford Theory for Modular Representations of Finite Groups
Devjani Basu
TL;DR
The paper extends Clifford theory to modular representations of finite groups by developing Nakayama-based tools, inertia groups, and induction/restriction analysis. It proves a decomposition theorem for $Res^G_N Ind^G_N \sigma$ and a Clifford correspondence via the inertia group, and then applies these results to $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$, presenting explicit polynomial realizations $Pol_k$ and $Pol_k(r)$ of their irreducibles, along with a $p=3$ example to illustrate the correspondence. The framework enables systematic study of modular representations of classical groups and their Lie-type relatives, with concrete constructions that link modular characters to representations via inertia and induction.
Abstract
Clifford theory establishes a relation between the representation theory of a finite group and its normal subgroups. In this paper, we establish the Clifford theory for the modular representations of finite groups. The proofs are based on an explicit analysis of the representation spaces and their decompositions. We also analyze the relation between the modular representations of $SL_2(\mathbb{F}_p)$ in defining characteristic, with that of $GL_2(\mathbb{F}_p)$ using the modular Clifford theory.