Group representation theory MOC

Group representation

A representation 𝔛 of a group 𝐺 is a linear group action on some vector space 𝑉 over 𝕂 rep2 i.e. a group homomorphism 𝔛 :𝐺 →GL⁡(𝑉), or equivalently a functor 𝔛 :𝐺 →𝖵𝖾𝖼𝗍𝕂 regarding groups as categories.¹ In particular, for any 𝑔,ℎ ∈𝐺

𝔛(𝑔)𝔛(ℎ)=𝔛(𝑔ℎ)

and 𝔛(1) =1𝑉.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 [[Module over a unital associative algebra|𝕂[𝐺]-module]], we often employ the abuse of terminology [[Module over a group|𝐺-module]] for (𝔛,𝑉) as a whole. To summarize, a representation is at once

• a group homomorphism 𝔛 :𝐺 →GL⁡(𝑉), which we use to emphasize the carrier space;

• a functor 𝔛 :𝐺 →𝖵𝖾𝖼𝗍𝕂, 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

• deg⁡𝔛 =dim𝕂⁡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 ⟨ ⋅| ⋅⟩ 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

1. Every group has a trivial (in general not faithful) representation Γ𝑇 : ⋅ ↦𝟙.
2. 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 𝖵𝖾𝖼𝗍𝕂, or equivalently as a module over a group ring. These yield two possible generalizations of representation.

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Footnotes

1. We will use both notations depending on which perspective is being emphasized. ↩

2. 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. ↩