Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.4 REDUCIBILITY OF A REPRESENTATION

Now, for somek, consider a vectorwin the spaceWkspanned by{sinkφ,coskφ},

sayw=asinkφ+bcoskφ. Under a rotation byαabout the polar axis,asinkφ

becomesasink(φ+α), which can be written asacoskαsinkφ+asinkαcoskφ,i.e

as a linear combination of sinkφand coskφ; similarly coskφbecomes another

linear combination of the same two functions. The newly generated vectorw′,

whose column matrixw′is given byw′=D(α)w, therefore belongs toWkfor

anyαand we can conclude thatWkis an invariant irreducible two-dimensional

subspace ofV. It follows thatD(α) is reducible and that, since the result holds

for everyk, in its reduced formD(α) has an infinite series of identical 2×2 blocks

on its leading diagonal; each block will have the form
(
cosα −sinα
sinα cosα

)

.

We note that the particular casek= 0 is special, in that then sinkφ= 0 and

coskφ= 1, for allφ; consequently the first 2×2 block inD(α) is reducible further

and becomes two single-entry blocks.

A second illustration of the connection between the behaviour of vector spaces

under the actions of the elements of a group and the form of the matrix repre-

sentation of the group is provided by the vector space spanned by the spherical

harmonicsYm(θ, φ). This contains subspaces, corresponding to the different

values of, that are invariant under the actions of the elements of the full three-

dimensional rotation group; the corresponding matrices are block-diagonal, and

those entries that correspond to the part of the basis containingYm(θ, φ)forma

(2+1)×(2+ 1) block.

To illustrate further the irreps of a group, we return again to the groupGof

two-dimensional rotation and reflection symmetries of an equilateral triangle, or

equivalently the permutation groupS 3 ; this may be shown, using the methods of

section 29.7 below, to have three irreps. Firstly, we have already seen that the set

Mof six orthogonal 2×2 matrices given in section (28.3), equation (28.13), is

isomorphic toG. These matrices therefore form not only a representation ofG,

but a faithful one. It should be noticed that, althoughGcontains six elements,

the matrices are only 2×2. However, they contain no invariant 1×1 sub-block

(which for 2×2 matrices would require them all to be diagonal) and neither can

allthe matrices be made block-diagonal by thesamesimilarity transformation;

they therefore form a two-dimensional irrep ofG.

Secondly, as previously noted, every group has one (unfaithful) irrep in which

every element is represented by the 1×1 matrixI 1 , or, more simply, 1.

Thirdly an (unfaithful) irrep ofGis given by assignment of the one-dimensional

set of six ‘matrices’{ 1 , 1 , 1 ,− 1 ,− 1 ,− 1 }to the symmetry operations{I, R, R′,K,

L, M}respectively, or to the group elements{I, A, B, C, D, E}respectively; see

section 28.3. In terms of the permutation groupS 3 , 1 corresponds to even

permutations and−1 to odd permutations, ‘odd’ or ‘even’ referring to the number