To this point we have constructed the matrix representations the group elements of point groups such as C3v in the usual physical space (two dimensional in our case). These representations were based on the transformations of the coordinates of an arbitrary vector in a physical space due to physical operations on the vector. Mathematical solutions to physical problems, however, are represented by functions in function spaces whose dimensions are generally much greater than three. Thus to bring the group matrix representations that act on coordinates to bear on the solution of physical problems in terms of functions, we need one more "connection" between symmetry operators on coordinates and symmetry operators on functions. This connection is defined as follows.

Let f (r) be a function of a position vector r = (x, y) and D (g-1) be the matrix transformation associated with group element g Є G, such that (x, y) ^ (x’, y’) through

r = D-1 (g)r.

What we need is an algorithm that uses D-1 (g) to obtain a new function h(r) from f (r). To this end we define an operator Og as

Ogf (r) = f (r’ )= f (D-1 (g)r) = h(r).

That is, operator Og gives a new function h(r) from f (r) at r, while f is unchanged at Г. For example, let

f (x, y) = ax + by

and

a + b a — b, . .

Og(ax + by) = ax + by = x + y = ^, y)

For two group elements g1 and g2 in G, we obtain

Og1 f (r)= f (D—1 (g)r) = h(r)

Og2Og1 f (r) = Og2 (OgJ(r)) = Og2h(r) = h(D—1(g2)r) = f ([D—1(g1 )D—1 fe)]r) = f ([D(g2)D(g1)]—1r). Note: operator Og acts on the coordinates of function f and not on the argument of f.

Therefore

Og2 Og1 f (r)= f ((d—1(g1 )D—1 (g2))—1 r) = f (D—1 (g1)D—1 (g2)D—1(g1 )r) = f ((D(g2)D(g1))—1 r),

in words: the consecutive application of Og1 and Og2 is the same as the application of the transformation Og2gt belonging to group element g1 g2, and the operators Og, g Є G have the same multiplication table as G and any group isomorphic with G.