regular representation

The regular representation is the group looking at itself: take the group algebra — formal linear combinations of group elements — and let the group act on it by left multiplication. It is the single most important representation because it contains, secretly inside, every irreducible representation of the group at once, each appearing exactly as many times as its own dimension.

Formally, let G be a finite group and k a field. The (left) regular representation is the vector space k[G] with basis the elements of G, where g acts by sending the basis vector h to the basis vector gh, extended linearly. Its degree is |G|. Its character chi_reg is dramatic: chi_reg(e) = |G| and chi_reg(g) = 0 for every g not equal to the identity, since left multiplication by a nonidentity element fixes no basis vector and so has zero trace.

Over an algebraically closed field of characteristic not dividing |G|, Maschke and Schur combine to give the decomposition k[G] = direct sum over irreducibles V_i of (dim V_i) copies of V_i. Taking dimensions recovers the sum-of-squares identity sum (dim V_i)^2 = |G|. Thus the regular representation is a 'universal container': decomposing it is equivalent to listing all irreducibles with multiplicity equal to dimension.