Reducibility of a representation

REPRESENTATION THEORY
a representation in which all the matrices are unitary (see chapter 8) and so from
now on we will consider only unitary representations.
29.4 Reducibility of a representation
We 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
unit n × n matrices In , as two of its possible representations.
Consider two or more representations, D(1) , D(2) , . . . , D(N) , which may be
of different dimensions, of a group G. Now combine the matrices D(1) (X),
D(2) (X), . . . , D(N) (X) that correspond to element X of G into a larger blockdiagonal matrix:
0
(1)
D (X )
(2)
D (X )
D(X ) =
(29.9)
..
.
D
0
(N)
(X )
Then D = {D(X)} is the matrix representation of the group obtained by combining
the basis vectors of D(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 representation D to 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, the same matrix Q for each of the matrices D(X)) then we would say
that D is reducible and that each matrix D(X) can be written as the direct sum of
smaller representations:
D(X) = D(1) (X) ⊕ D(2) (X) ⊕ · · · ⊕ D(N) (X).
It may be that some or all of the matrices D(1) (X), D(2) (X), . . . , D(N) themselves
can be further reduced – i.e. written in block diagonal form. For example,
suppose that the representation D(1) , say, has a basis vector (x y z)T ; then, for
the symmetry group of an equilateral triangle, whilst x and y are mixed together
for at least one of the operations X, z is never changed. In this case the 3 × 3
representative matrix D(1) (X) can itself be written in block diagonal form as a
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29.4 REDUCIBILITY OF A REPRESENTATION
2 × 2 matrix and a 1 × 1 matrix. The direct-sum matrix D(X) can now be written
a
b
c
d
0
1
(2)
D(X ) =
D (X )
(29.10)
..
.
D
0
(N)
(X )
but the first two blocks can be reduced no further.
When all the other representations D(2) (X), . . . have been similarly treated,
what remains is said to be irreducible and has the characteristic of being block
diagonal, with blocks that individually cannot be reduced further. The blocks are
known as the irreducible representations of G, often abbreviated to the irreps of
(i)
G, and we denote them by D̂ . 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 of G.
Any representation can be written as a linear combination of irreps.
If, however, the initial choice u of basis vector for the representation D is
arbitrary, as it is in general, then it is unlikely that the matrices D(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 transformation Q to a new basis vector
uQ , as described in section 29.3.
(i)
In any particular representation D, each constituent irrep D̂ 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 of D itself. Let us say that
(i)
D̂ appears mi times. The general expansion of D is then written
D = m1 D̂
(1)
⊕ m2 D̂
(2)
⊕ · · · ⊕ mN D̂
(N)
,
(29.11)
where if G is finite so is N.
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
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REPRESENTATION THEORY
of a particular group of symmetry operations has been brought to irreducible
form, the implications are as follows.
(i) Those components of the basis vector that correspond to rows in the
representation matrices with a single-entry block, i.e. a 1 × 1 block, are
unchanged by the operations of the group. Such a coordinate or function
is said to transform according to a one-dimensional irrep of G. In the
example given in (29.10), that the entry on the third row forms a 1 × 1
block implies that the third entry in the basis vector (x y z · · · )T ,
namely z, is invariant under the two-dimensional symmetry operations on
an equilateral triangle in the xy-plane.
(ii) If, in any of the g matrices of the representation, the largest-sized block
located on the row or column corresponding to a particular coordinate
(or function) in the basis vector is n × n, then that coordinate (or function)
is mixed by the symmetry operations with n − 1 others and is said to
transform according to an n-dimensional irrep of G. Thus in the matrix
(29.10), x is the first entry in the complete basis vector; the first row of
the matrix contains two non-zero entries, as does the first column, and so
x is part of a two-component basis vector whose components are mixed
by the symmetry operations of G. The other component is y.
The result (29.11) may also be formulated in terms of the more abstract notion
of vector spaces (chapter 8). The set of g matrices that forms an n-dimensional
representation D of the group G can be thought of as acting on column matrices
corresponding to vectors in an n-dimensional vector space V spanned by the basis
functions of the representation. If there exists a proper subspace W of V , such
that if a vector whose column matrix is w belongs to W then the vector whose
column matrix is D(X)w also belongs to W , for all X belonging to G, then it
follows that D is reducible. We say that the subspace W is invariant under the
actions of the elements of G. With D unitary, the orthogonal complement W⊥ of
W , i.e. the vector space V remaining when the subspace W has been removed, is
also invariant, and all the matrices D(X) split into two blocks acting separately
on W and W⊥ . Both W and W⊥ may contain further invariant subspaces, in
which case the matrices will be split still further.
As a concrete example of this approach, consider in plane polar coordinates
ρ, φ the effect of rotations about the polar axis on the infinite-dimensional vector
space V of all functions of φ that satisfy the Dirichlet conditions for expansion
as a Fourier series (see section 12.1). We take as our basis functions the set
{sin mφ, cos mφ} for integer values m = 0, 1, 2, . . . ; this is an infinite-dimensional
representation (n = ∞) and, since a rotation about the polar axis can be through
any angle α (0 ≤ α < 2π), the group G is a subgroup of the continuous rotation
group and has its order g formally equal to infinity.
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29.4 REDUCIBILITY OF A REPRESENTATION
Now, for some k, consider a vector w in the space Wk spanned by {sin kφ, cos kφ},
say w = a sin kφ + b cos kφ. Under a rotation by α about the polar axis, a sin kφ
becomes a sin k(φ + α), which can be written as a cos kα sin kφ + a sin kα cos kφ, i.e
as a linear combination of sin kφ and cos kφ; similarly cos kφ becomes another
linear combination of the same two functions. The newly generated vector w ,
whose column matrix w is given by w = D(α)w, therefore belongs to Wk for
any α and we can conclude that Wk is an invariant irreducible two-dimensional
subspace of V . It follows that D(α) is reducible and that, since the result holds
for every k, in its reduced form D(α) has an infinite series of identical 2 × 2 blocks
on its leading diagonal; each block will have the form
cos α − sin α
.
sin α cos α
We note that the particular case k = 0 is special, in that then sin kφ = 0 and
cos kφ = 1, for all φ; consequently the first 2 × 2 block in D(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 representation of the group is provided by the vector space spanned by the spherical
harmonics Ym (θ, φ). This contains subspaces, corresponding to the different
values of , that are invariant under the actions of the elements of the full threedimensional rotation group; the corresponding matrices are block-diagonal, and
those entries that correspond to the part of the basis containing Ym (θ, φ) form a
(2 + 1) × (2 + 1) block.
To illustrate further the irreps of a group, we return again to the group G of
two-dimensional rotation and reflection symmetries of an equilateral triangle, or
equivalently the permutation group S3 ; this may be shown, using the methods of
section 29.7 below, to have three irreps. Firstly, we have already seen that the set
M of six orthogonal 2 × 2 matrices given in section (28.3), equation (28.13), is
isomorphic to G. These matrices therefore form not only a representation of G,
but a faithful one. It should be noticed that, although G contains 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
all the matrices be made block-diagonal by the same similarity transformation;
they therefore form a two-dimensional irrep of G.
Secondly, as previously noted, every group has one (unfaithful) irrep in which
every element is represented by the 1 × 1 matrix I1 , or, more simply, 1.
Thirdly an (unfaithful) irrep of G is 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 group S3 , 1 corresponds to even
permutations and −1 to odd permutations, ‘odd’ or ‘even’ referring to the number
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