rightregular

In group theory, the right regular representation of a group G is the permutation representation of G on the underlying set G given by right multiplication. For each g in G, define R_g: G → G by R_g(x) = xg. The map g ↦ R_g is a homomorphism from G into the symmetric group Sym(G), providing a representation of G.

The action described by R is simply transitive and free: for any x in G, the orbit of x under the action is the whole group, and the stabilizer of x is the trivial group {e}. Consequently, the right regular action is a regular (or simply transitive) action.

If G is finite, the right regular representation has dimension equal to the order of G when viewed as a representation on the vector space with basis elements indexed by G (the group algebra CG). Concretely, R_g permutes the basis by sending basis element corresponding to x to the one corresponding to xg. It is the permutation representation afforded by the action of G on itself by right multiplication. The character of the right regular representation satisfies χ_R(g) = |G| if g = e and χ_R(g) = 0 otherwise.

Relation to the left regular representation: the left regular representation L_g(x) = gx acts on G by left multiplication. The two representations are isomorphic (via the inversion map x ↦ x^{-1}): L_g is intertwined with R_g^{-1} by x ↦ x^{-1}, so they have the same structure up to this natural identification.

Applications of the right regular representation include the study of representation theory of finite groups, Fourier analysis on finite groups, and the examination of permutation representations arising from group actions. The term “right-regular” emphasizes the direction of the action on G.