Regular representation

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 character of the regular representation takes the value $$|G|$$ 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:


 * Splitting implies characters form a basis for space of class functions
 * Sum of squares of degrees of irreducible representations equals order of group