Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.4 REDUCIBILITY OF A REPRESENTATION

2 ×2matrixanda1×1 matrix. The direct-sum matrixD(X) can now be written

a b

cd

1

D

(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 representationDis

arbitrary, 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 any

number 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 in

somewhat different language. When the set of matrices that forms a representation