Regular representation

In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.

One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation.

For a finite group G, the left regular representation λ (over a field K) is a linear representation on the K-vector space V freely generated by the elements of G, i. e. they can be identified with a basis of V. Given g ∈ G, λ(g) is the linear map determined by its action on the basis by left translation by g, i.e.

For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given g ∈ G, ρ(g) is the linear map on V determined by its action on the basis by right translation by g⁻¹, i.e.

Alternatively, these representations can be defined on the K-vector space W of all functions G → K. It is in this form that the regular representation is generalized to topological groups such as Lie groups.

The specific definition in terms of W is as follows. Given a function f : G → K and an element g ∈ G,

and

To say that G acts on itself by multiplication is tautological. If we consider this action as a permutation representation it is characterised as having a single orbit and stabilizer the identity subgroup {e} of G. The regular representation of G, for a given field K, is the linear representation made by taking this permutation representation as a set of basis vectors of a vector space over K. The significance is that while the permutation representation doesn't decompose - it is transitive - the regular representation in general breaks up into smaller representations. For example, if G is a finite group and K is the complex number field, the regular representation decomposes as a direct sum of irreducible representations, with each irreducible representation appearing in the decomposition with multiplicity its dimension. The number of these irreducibles is equal to the number of conjugacy classes of G.

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