{c}

<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.<Modular Representations of Finite Groups of Lie Type by James E. Humphreys>[Chapter 1, especially §§1.1 and 1.3]{ref}

A small example illustrates why the <characteristic of a field> matters in <representation theory>. Over the <finite field> $\mathbb F_2$, the <matrix>
$$
A=\begin{pmatrix}1&1\\0&1\end{pmatrix}
$$
satisfies $A^2=I$ 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.


 James E. Humphreys (source code)