REPRESENTATION THEORY
a representation in which all the matrices are unitary (see chapter 8) and so from
now on we will consider onlyunitary representations.
29.4 Reducibility of a representationWe have seen already that it is possible to have more than one representation
of any particular group. For example, the group{ 1 ,i,− 1 ,−i}under ordinary
multiplication has been shown to have a set of 2×2 matrices, and a set of four
unitn×nmatricesIn, as two of its possible representations.
Consider two or more representations,D(1), D(2), ...,D(N), which may beof different dimensions, of a groupG. Now combine the matricesD(1)(X),
D(2)(X), ...,D(N)(X) that correspond to elementXofGinto a largerblock-
diagonalmatrix:
D(2)
(X)D(N)(X)...
D(X) =0
0
D(1)
(X)(29.9)ThenD={D(X)}is the matrix representation of the group obtained by combining
the basis vectors ofD(1),D(2), ...,D(N)into one larger basis vector. If, knowingly or
unknowingly, we had started with this larger basis vector and found the matrices
of the representationDto have the form shown in (29.9), or to have a form
that can be transformed into this by a similarity transformation (29.5) (using,
of course, thesamematrixQfor each of the matricesD(X)) then we would say
thatDisreducibleand that each matrixD(X) can be written as thedirect sumof
smaller representations:
D(X)=D(1)(X)⊕D(2)(X)⊕···⊕D(N)(X).It may be that some or all of the matricesD(1)(X),D(2)(X), ...,D(N)themselvescan be further reduced – i.e. written in block diagonal form. For example,
suppose that the representationD(1), say, has a basis vector (xyz)T; then, for
the symmetry group of an equilateral triangle, whilstxandyare mixed together
for at least one of the operationsX,zis never changed. In this case the 3× 3
representative matrixD(1)(X) can itself be written in block diagonal form as a