## Invariant subspaces and regular representations

A common approach to the solution of physical problems is harmonic analysis, where a solution to the problem is sought in terms of functions that span the solution space. If the problem exhibits some symmetry, we would expect this symmetry to be reflected in the solution for this particular problem. Intuitively we would expect therefore the solution to belong to a subspace of the general solution space, and that the subspace be invariant under the symmetry operations exhibited by the problem.

As an illustration of this notion, we assume the problem has the symmetry of the cyclic permutation group C3 = {E, C3, } that was discussed previously. Let fe(r) be an arbitrary

function that allows the operation of the operators in the group C3 as discussed above. The action of each operator on fE defines a new function that, is

OEfE = fE OC3fE = fC3 OC2fE = fC.

Based on this and the group multiplication table we get relations such as

OC3 fC3 = OC3 OC3fE = OC2fE = fC2,

etc. These observations can be summarized in a table: From that table we can construct matrix (permutation) representations of the operators OE, OC3 , OC2 as for example D(C3 ) = (2,3,1).

 fE fC3 fc2 fE fE fC3 fc2 C3 fC3 fC3 fc2 fE fC2 fC2 fE fC3

This procedure gives the so-called regular representation for the group C3 as

Oe = (1,2,3); OC3 = (2,3,1); Oq = (3,1,2). (2.9)

The matrices, in general, satisfy the group multiplication table, and are characterized by only the one integer one in each column, the rest zeros, and the dimension of the matrix equals to the number of elements in the group. The functions fe, fc3, fc2 that generate the regular representation, span the invariant subspace. They are not necessarily linearly independent basis functions.