Group representation theory MOC

Group representation

A representation of a group is a linear group action on some vector space over #m/def/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, we often employ the abuse of terminology -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

• 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 .
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.

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