8 2. Basic notions of representation theory

The graphs listed in the theorem are called (simply laced) Dyn-

kin diagrams. These graphs arise in a multitude of classification

problems in mathematics, such as the classification of simple Lie al-

gebras, singularities, platonic solids, reflection groups, etc. In fact, if

we needed to make contact with an alien civilization and show them

how sophisticated our civilization is, perhaps showing them Dynkin

diagrams would be the best choice!

As a final example consider the representation theory of finite

groups, which is one of the most fascinating chapters of represen-

tation theory. In this theory, one considers representations of the

group algebra A = C[G] of a finite group G — the algebra with basis

ag,g ∈ G, and multiplication law agah = agh. We will show that any

finite dimensional representation of A is a direct sum of irreducible

representations, i.e., the notions of an irreducible and indecompos-

able representation are the same for A (Maschke’s theorem). Another

striking result discussed below is the Frobenius divisibility theorem:

the dimension of any irreducible representation of A divides the or-

der of G. Finally, we will show how to use the representation theory

of finite groups to prove Burnside’s theorem: any finite group of or-

der paqb, where p, q are primes, is solvable. Note that this theorem

does not mention representations, which are used only in its proof; a

purely group-theoretical proof of this theorem (not using representa-

tions) exists but is much more diﬃcult!

2.2. Algebras

Let us now begin a systematic discussion of representation theory.

Let k be a field. Unless stated otherwise, we will always assume

that k is algebraically closed, i.e., any nonconstant polynomial with

coeﬃcients in k has a root in k. The main example is the field of

complex numbers C, but we will also consider fields of characteristic

p, such as the algebraic closure Fp of the finite field Fp of p elements.

Definition 2.2.1. An associative algebra over k is a vector space

A over k together with a bilinear map A × A → A, (a, b) → ab, such

that (ab)c = a(bc).