jaj•a•person's notes

Group representation theory MOC

Group representation

A representation of a group is a linear group action on some vector space over rep2 i.e. a group homomorphism , or equivalently a functor regarding groups as categories.1 In particular, for any

and .2 Since a representation of over uniquely determines a representation of the group ring and vice versa, the latter being equivalent to a [[Module over a unital associative algebra|-module]], we often employ the abuse of terminology [[Module over a group|-module]] for as a whole. To summarize, a representation is at once

  • a group homomorphism , which we use to emphasize the carrier space;

  • a functor , which we use to emphasize the ground field;

  • a module over , which we use to consider the aggregate as a single object.

Additional terminology

  • is the degree of the representation.
  • The vector space is said to carry the representation , 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

Carrier space symmetry

Properties

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

tidy | en | sembr

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. ↩

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