Why represent a group with matrices at all?

Linear algebra is far more computable than abstract group theory, and characters reduce a representation to a single class function. Frobenius's character theory let mathematicians prove deep results, such as Burnside's theorem on groups of order divisible by only two primes, that were otherwise inaccessible.

What does it mean for a representation to be irreducible?

An irreducible representation has no proper nonzero subspace preserved by every group element; it is a building block. Maschke's theorem says that in good characteristic every representation is a direct sum of these blocks.

Group Representation

A group representation realizes the elements of a group as invertible linear transformations of a vector space, translating group theory into linear algebra and exposing structure through characters.

Najít téma v PaperMindJiž brzyFind papers & topics

Tools & resources

Stáhnout prezentaci

Learn & explore

VideoJiž brzy

Definition

A representation of a group G on a vector space V is a homomorphism from G to the group of invertible linear operators on V, equivalently a module over the group algebra of G.

Scope

This topic covers representations and their equivalence, irreducible representations, Maschke's theorem on complete reducibility, Schur's lemma, characters and orthogonality relations, and the decomposition of representations over fields of characteristic zero. It is the gateway to the representation theory of finite groups.

Core questions

How can a group be modeled by matrices acting on a vector space?
When does a representation decompose into irreducible pieces?
What information about a representation is captured by its character?
How do orthogonality relations classify the irreducible representations of a finite group?

Key theories

Maschke's theorem: Over a field whose characteristic does not divide the group order, every representation of a finite group is completely reducible, decomposing as a direct sum of irreducible representations.
Schur's lemma: Any homomorphism between irreducible representations is either zero or an isomorphism, and over an algebraically closed field the endomorphisms of an irreducible representation are scalars, the cornerstone of character theory.
Character orthogonality relations: The characters of the irreducible complex representations of a finite group form an orthonormal basis for the space of class functions, so the number of irreducibles equals the number of conjugacy classes and every representation is determined by its character.

Clinical relevance

Representation theory makes finite groups computable through linear algebra and is indispensable in quantum mechanics and spectroscopy (symmetry-adapted bases and selection rules), in crystallography, and in the analysis of symmetry in physics, as well as in number theory through the representations attached to Galois groups.

History

Frobenius introduced characters and representations of finite groups in the 1890s, and Schur, Burnside, and Weyl developed the theory into a powerful structural tool. Maschke's theorem and the orthogonality relations gave the subject the form taught today and connected it to the physics of symmetry.

Key figures

Georg Frobenius
Issai Schur
William Burnside
Hermann Weyl

Seminal works

serre1977
dummit2004
lang2002

Frequently asked questions

Why represent a group with matrices at all?: Linear algebra is far more computable than abstract group theory, and characters reduce a representation to a single class function. Frobenius's character theory let mathematicians prove deep results, such as Burnside's theorem on groups of order divisible by only two primes, that were otherwise inaccessible.
What does it mean for a representation to be irreducible?: An irreducible representation has no proper nonzero subspace preserved by every group element; it is a building block. Maschke's theorem says that in good characteristic every representation is a direct sum of these blocks.