Representation theory of finite groups

Builds on

Required prerequisites, all the way down: 25

You will learn to

• Decompose representations of small finite groups into irreducibles using characters.
• Prove Maschke's theorem and Schur's lemma and derive the orthogonality relations.
• Construct the character tables of S₃, S₄, D₄ and Q₈, and use Frobenius reciprocity for induced representations.

Definition

• A representation of G on a vector space V is a homomorphism ρ : G → GL(V).
• A representation is irreducible if it is non-zero and has no invariant subspaces other than 0 and V.
• The character of ρ is the function χ(g) = tr ρ(g).

Theorem

• Maschke: every finite-dimensional representation of a finite group over a field whose characteristic does not divide |G| is a direct sum of irreducible representations.
• The number of irreducible complex representations of G equals the number of conjugacy classes of G.
• The squares of the dimensions of the irreducible complex representations sum to |G|.
• Non-isomorphic groups can have the same character table: D₄ and Q₈ do.

Common misconceptions

• Assuming complete reducibility when the characteristic divides |G|.
• Thinking the character table determines the group up to isomorphism.

Leads to

Optional discoveries

• Functors A representation of G is a functor from the one-object category BG to the category of vector spaces, and intertwining maps are natural transformations between such functors.
• Mathematical foundations of quantum mechanics Symmetry groups act on quantum state spaces, and irreducible representations organise energy levels and quantum numbers.
• ← Linear algebra
• ← Enumerative combinatorics

Learning pathways

• Graduate preparation
• Research preparation

Further reading

• Jean-Pierre Serre, Linear Representations of Finite Groups (1977)
• William Fulton and Joe Harris, Representation Theory: A First Course (1991)