 James E. Humphreys

James E. Humphreys wrote influential mathematical expositions on Lie theory and representation theory. His book Modular Representations of Finite Groups of Lie Type by James E. Humphreys develops modular representation theory for finite groups of Lie type, relating finite groups to algebraic groups over finite fields. The opening chapter describes constructions using rational points and fixed points of appropriate endomorphisms.^[ref]

A small example illustrates why the characteristic of a field matters in representation theory. Over the finite field ${\mathbb{F}}_2$, the matrixsatisfies $A^2=I$ but $A\ne I$, so it represents the generator of a group of order two. Its fixed vectors form only the line spanned by $(1,0)$. That line has no invariant one-dimensional complement: over ${\mathbb{F}}_2$, an invertible action on a line must be the identity, so two invariant lines would force $A=I$. The example shows concretely how a representation can fail to split into smaller pieces when the field characteristic divides the group order. This is part of the mathematical setting that Humphreys's expositions make accessible.