Representations of groups

Warning

This page is currently under construction. Consider everything here tentative until this warning is removed.

Types of representations

In this note we will stick to representations of groups, but in the back of our minds we should consider the extent to which we can extend these concepts to other structures.

Matrix representations over a field

Definition of matrix representation of a group

Let $G$ be a group and $F$ be a field. A matrix representation of $G$ over $F$ is a group morphism

ϕ : G \to {GL}_{n} (F),

where ${GL}_{n} (F)$ is the group of invertible $n \times n$ matrices with entries in $F$ .

In other words, a matrix representation of $G$ assigns to each group element $g \in G$ an invertible $n \times n$ matrix $ϕ (g)$ , in a manner compatible with the group structure; i.e., such that $ϕ (g_{1} g_{2}) = ϕ (g_{1}) ϕ (g_{2})$ for all $g_{1}, g_{2} \in G$ .

Example

(coming soon)

Vector space representations

Definition of vector space representation of a group

Let $G$ be a group and $F$ be a field. A vector space representation of $G$ over $F$ is a group morphism

ϕ : G \to GL (V),

where $V$ is an $F$ -vector space and $GL (V)$ is the group of invertible $F$ -linear endomorphisms of $V$ .

(mention connection between this and matrix representations)

Example

(coming soon)

Permutation representations

Definition of a matrix representation of a group

Let $G$ be a group. A permutation representation of $G$ is a group morphism

ϕ : G \to S_{X},

where $X$ is a set and $S_{X}$ is the group of permutations (i.e., self-bijections) of $X$ .

Example

(include maybe a dihedral group acting on some geometric feature of the polygon)

Representations in your favorite category

Definition of a matrix representation of a group

Let $G$ be a group and $C$ be a category. A representation of $G$ in $C$ is a group morphism

ϕ : G \to {Aut}_{C} (c),

where $c$ is an object of $C$ and ${Aut}_{C} (c)$ is the group of automorphisms of $c$ in $C$ .

Notice that this final definition subsumes the previous three. Indeed, observe:

If $C = {Mat}_{F}$ is the category of matrices with entries in $F$ , then each object in $C$ corresponds to a positive integer $n$ . The arrows $n \to n$ correspond to $n \times n$ matrices with entries in $F$ , and the automorphisms of $n$ correspond to invertible $n \times n$ matrices. Thus, ${Aut}_{{Mat}_{F}} (n) = {GL}_{n} (F)$ .
If $C = {Vec}_{F}$ is the category of $F$ -vector spaces, then each object in $C$ corresponds to an $F$ -vector space $V$ . The arrows $V \to V$ correspond to $F$ -linear transformations, and the automorphisms of $V$ correspond to invertible $F$ -linear endomorphisms of $V$ . Thus, ${Aut}_{{Vec}_{F}} (V) = GL (V)$ .
If $C = Set$ is the category of sets, then each object in $C$ corresponds to a set $X$ . The arrows $X \to X$ correspond to set maps, and the automorphisms of $X$ correspond to bijective set maps $X \to X$ ; i.e., permutations of $X$ . Thus, ${Aut}_{Set} (X) = S_{X}$ .

Representations as functors

Suggested next note

Similar representations