29.4 REDUCIBILITY OF A REPRESENTATION
2 ×2matrixanda1×1 matrix. The direct-sum matrixD(X) can now be written
a bcd
1D(2)
(X)D(N)
(X)...
D(X) =0
0
(29.10)but the first two blocks can be reduced no further.
When all the other representationsD(2)(X),...have been similarly treated,what remains is said to beirreducibleand has the characteristic of being block
diagonal, with blocks that individually cannot be reduced further. The blocks are
known as theirreducible representations ofG, often abbreviated to theirreps of
G, and we denote them byDˆ
(i). They form the building blocks of representation
theory, and it is their properties that are used to analyse any given physical
situation which is invariant under the operations that form the elements ofG.
Any representation can be written as a linear combination of irreps.
If, however, the initial choiceuof basis vector for the representationDisarbitrary, as it is in general, then it is unlikely that the matricesD(X) will
assume obviously block diagonal forms (it should be noted, though, that since
the matrices are square, even a matrix with non-zero entries only in the extreme
top right and bottom left positions is technically block diagonal). In general, it
will be possible to reduce them to block diagonal matrices with more than one
block; this reduction corresponds to a transformationQto a new basis vector
uQ, as described in section 29.3.
In any particular representationD, each constituent irrepDˆ(i)
may appear anynumber of times, or not at all, subject to the obvious restriction that the sum of
all the irrep dimensions must add up to the dimension ofDitself. Let us say that
Dˆ(i)appearsmitimes. The general expansion ofDis then written
D=m 1 Dˆ(1)
⊕m 2 Dˆ(2)
⊕···⊕mNDˆ(N)
, (29.11)where ifGis finite so isN.
This is such an important result that we shall now restate the situation insomewhat different language. When the set of matrices that forms a representation