Group representation

A representation $𝔛$ of a group $𝐺$ is a linear group action on some vector space $𝑉$ over $𝕂$ #m/def/rep2 i.e. a group homomorphism $𝔛 : 𝐺 \to G L (𝑉)$ , or equivalently a functor $𝔛 : 𝐺 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ regarding Groups as groupoids.¹ In particular, for any $𝑔, ℎ \in 𝐺$

𝔛 (𝑔) 𝔛 (ℎ) = 𝔛 (𝑔 ℎ)

and $𝔛 (1) = 1 𝑉$ .² Since a representation of $𝐺$ over $𝕂$ uniquely determines a representation of the group ring $𝕂 [𝐺]$ and vice versa, the latter being equivalent to a $𝕂 [𝐺]$ -module, we often employ the abuse of terminology $𝐺$ -module for $(𝔛, 𝑉)$ as a whole. To summarize, a representation is at once

a group homomorphism $𝔛 : 𝐺 \to G L (𝑉)$ , which we use to emphasize the carrier space;
a functor $𝔛 : 𝐺 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ , which we use to emphasize the ground field;
a module $𝑉$ over $𝕂 [𝐺]$ , which we use to consider the aggregate as a single object.

Additional terminology

$d e g 𝔛 = d i m 𝕂 𝑉$ is the degree of the representation.
The vector space $𝑉$ is said to carry the representation $𝔛$ , and is also called the carrier space.
In these notes, if the carrier space is an Inner product space it will usually use the linear-second $⟨ \cdot | \cdot ⟩$ convention, signalled by the bar.
With a fixed basis, we can use a Matrix representation.

Types of representation

A Faithful representation is injective
A Full representation is surjective
A Fully faithful representation is bijective
Representations may also be classified by reducibility.

Carrier space symmetry

A Unitary representation is unitary for every group element.
A Symplectic group representation is symplectic for every group element.

Properties

Every group has a trivial (in general not faithful) representation $Γ 𝑇 : \cdot \mapsto 𝟙$ .
A non-trivial non-faithful representation implies a non-trivial normal subgroup

Generalizations

A representation may be viewed as a Functor from a single-object Groupoid to $𝖵 𝖾 𝖼 𝗍 𝕂$ , or equivalently as a module over a group ring. These yield two possible generalizations of representation.

#state/tidy | #lang/en | #SemBr

We will use both notations depending on which perspective is being emphasized. ↩
2023, Groups and representations, p. 20 Since Every finite complex representation of a compact group is equivalent to a unitary representation, it is common to only consider unitary representations. ↩