Regular representation

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Definition

As a group action

The regular representation of a group G over a field K is the permutation representation arising from the regular group action, i.e, the action of the group on itself as a set by left multiplication (or right multiplication if we're using a right action convention).

The regular representation of G is therefore a representation over a vector space with basis indexed by the elements of G. In particular, the vector space is a |G|-dimensional vector space. In particular, it is finite-dimensional if the group is a finite group.

As a module over the group ring

The regular representation of a group G over a field K corresponds to viewing K[G] as a module over itself with the usual left multiplication.

Facts

The regular representation is also used in proving various things such as: