As a group action
The regular representation of a group over a field 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 is therefore a representation over a vector space with basis indexed by the elements of . In particular, the vector space is a -dimensional vector space. In particular, it is finite-dimensional if the group is a finite group.
As a module over the group ring
- The character of the regular representation takes the value at the identity element and zero elsewhere. This is because the character of a permutation matrix is the number of fixed points, and multiplication by a non-identity element has no fixed points.
- Regular representation over splitting field has multiplicity of each irreducible representation equal to its degree
The regular representation is also used in proving various things such as: