Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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24,
_
Group Representation Theory
Group action is extremely important in quantum mechanics. Suppose the Hamil-
tonian of a quantum system is invariant under a symmetry transformation of its
independent
parameters
such as position,
momentum,
andtime. This
invariance
will show up as certain properties of the solutions of the Schrodinger equation.
Moreover, the very act of labeling quantum-mechanical states often involves
groups and their actions. For example, labeling atomic states by eigenvalues
of
angular momentum assumes invariance of the Hamiltonianunder the action of the
rotation group (see Chapter 27) on the Hilbert space of the quantum-mechanical
system
under
consideration.
24.1 Definitions and Examples
In
the languageof group theory, we have the following situation.
Put
all the param-
eters Xl, ,
x
p
of the Hamiltonian Htogether to form a space, say
IRP,
and write
H= H(XI, , x
p
)
=H(x). A
group
of
symmetry
ofH is a group G whose action
on
IRP
leaves Hunchanged,' i.e., H(x· g) = H(x). For example, aone-dimensional
li
2
d
2
harmonic
oscillator, with H =
--
2 + imw2x2, has, among otherthings,
2mdx
parity P (defined by
Px
=
-x)
as a symmetry. Thus, the group G ={e, P} is a
group
of
symmetry of H.
The Hamiltonian Hof a quantum-mechanical system is an operatorin aHilbert
space, such as
.c,2(IR
3),
the space of square-integrable functions. The important
question is, What is the proper way
of
transporting the action of G from
IRP
1It willbecomeclear
shortly
thatthe
appropriate
direction
fortheactionis fromthe
right.
674
24.
GROUP
REPRESENTATION
THEORY
representation;
carrier
space
and
dimension
a!a
representation;
taith!ul
and
identity
representation
to ,(,2(Jll3)? This is a relevant question because the solutions
of
the Schrodinger
equation are, in general, functions of the parameters of the Hamiltonian, and as
soch will be affected by the symmetry operation on the Hamiltonian. The answer
is provided in the following definition.
24.1.1. Definition.
Let
G be a group and
~
a Hilbert space. A representation
of
G on
~
is a homomorphism T : G --* GL
(~).
The representation isfaithful if
the homomorphism is 1-1. We often denote
T(g)
by T
g
•
~
is called the carrier
space
of
T. The trivial homomorphism T : G --* {1} is also called the identity
representation. The dimension
of~
is called the dimension
of
the representation
T.
We do not want to distingnish between representations that differ only by
isomorphic vector spaces, because otherwise we can generate an
infinite
set
of
representationsthat are trivially relatedto one another. Avector spaceisomorphism
f:
~
--*~'
induces a group isomorphism
</J
:
GL(~
--*
GL(~')
defined by
for T
E
GL(~).
equivalent
representations
This motivates the following definition.
24.1.2. Definition.
Two representations T : G --*
GL(~)
and
T'
: G --*
G
L(~')
are called equivalent ifthere exists an isomorphism f :
~
--*
~'
such
thatT~
= f 0 T
g
0
r:'
for all g E G.
24.1.3. Box. Any representation T : G --*
GL(~)
defines an action
of
the
group
G on the Hilbert space
~
by
<I>(g,
10))
sa T
g
10).
As we saw in Chapters 2 and 3, the transformation
of
an operator A under
T
g
would have to be defined by
TgA(Tg)-I.
For a Hamiltonian with a group of
symmetry G, this leads to the identity
Tg[H(x)](Tg)-l =
H(x·
g).
Similarly, the action of the group on a vector (function) in ,(,2(Ill3) is defined by
(T
g
1/r
)(x) sa 1/r(x, g),
(24.1)
where the parentheses around Tg
1/r
designate it as a new function. One can show
that
if
G acts on the independentvariables
of
a function on the right as in Equation
(24.1), then thevector space of such functions isthe carrierspace
of
arepresentation
of G. 10fact,
Energyeigenstates
canbe
labeled
by
eigenvalues
ofthe
symmetryoperators
aswell.
matrix
representations
24.1
DEFINITIONS
AND
EXAMPLES
675
where
we have defined
the
new
function
<p
by the
last
equality.
Now
note
that
It
follows from the last
two
equations
that
Tg",1/r = T
gIT
g,1/r.
Siuce this
holds
for arbitrary
1/r,
we
must
have
T
gIg,
= TglT
ez
i.e., that T is a
representation.
When
the action
of
a
group
is "naturally"
from
the left,
such
as the
action
of
a matrix on a
colmon
vector, we replace x . g
with
g
-1
. x.
The
reader
can
check
that T : G --*
GL(j{),
given
by
T
g1/r(x)
=
1/r(g-l
.
x),
is iudeed a
representation.
24.1.4.
Example.
Let the Hamiltonian of the time-iudependent Schrodinger equation
H
11/r)
= E
10/)
be iuvariantunder the actionof a group G. This means that
TgHT
g
l
= H
=>
[H,Tgl = 0,
i.e., that Hand Tg are simultaneouslydiagonalizable (Theorem
4.4.15).
It
followsthat we
canchoosethe energyeigenstatestobe eigenstates ofTg as well, and we canlabel thestates
not only by the energy "quantumnumbers"--eigenvalues
of
H-but
also bythe eigenvalues
of Tg. Forexample,
if
theHamiltonian is invariantunder the action of parity P, thenwe can
choosethestalestobeeven,correspondingtoparityeigenvalueof +1,orodd,corresponding
toparity eigenvalueof
-1.
Similarly,if G istherotation group,thenthe stalescanbelabeled
by the eigenvalues
of
the rotation operators, which are, as we shall see, equivalent to the
angular momentum operators discussed in Chapter 12.
In crystallography and solid-state physics, the Hamiltonian of an (infinite) lattice is
invariant undertranslation by an integermultiple
of
eachso-calledprimitive lattice transla-
tion, the three noncoplanar vectors that define a primitive cell
of
the crystal. The preceding
argument shows that the energy eigenstates can
betaken to be the eigenstates
of
the trans-
lation operator as well.
IIlII
It
is
common
to choose a basis
and
representall Tg'S in terms
of
matrices.
Then
one
gets a
matrix
representation
of
the group
G.
24.1.5.
Example.
Coosider the action of the 2D rotatioo group
80(2)
(rotation about
the z-axis) on
JR.3:
x' =
xcos9
-
ysin9,
r'=Rz(B)r
=>
y'=xsinB+ycosB,
z'w e.
For a Hilbert space, also choose
JR.3.
Define the homomorphism T : G --+
GL(J[)
to be
the identitymap, so that T(Rz(B)) eaTe = Rz(B). TbeoperatorTe transformsthestandard
basis vectors of:J{ as
Teel =
Te(l,
0, 0) = (cosB, sinB, 0) = cosBel + sinBe2 + Oe3'
Tee2 =
Te(O,
1,0)
=
(-
siuB,coss, 0) = - siuBel + cosBe2 +
Oe3,
Tee3 = Te(O,0, I) =
(0,0,
I) = Oel +
Oe2
+ e3.
676 24.
GROUP
REPRESENTATION
THEORY
It follows
that
thematrix representation of
80(2)
in the
standard
basisof
j{
is
(
cose
TO
=
s~e
-
sioe
0)
case
O.
o 1
Note
that
80(2)
is an
infinite
group;
its
cardinality
is determined by the "number" of
8~.
g
24.1.6. Example. Let 83 act on
lll.3
on the right by shnflling components:
For
the carrier space,
chooseR
3
as well.
LetT : 83
-+
GL(lR3) begivenas follows: T(n:)
Xl
x;r(l)
is the
matrix
that
takes
the
column
vectorx =
(xz)
to (X;r(2)
).
As a specific
illustration,
X3 X;r(3)
consider"
= (j j
~)
and write Tn for T
(x).
Then
Tn
(e,)
=T
n
(l , 0, 0) =
(1,0,0)."
= (0,
1,0)
= e2,
Tn(ez)
= Tn (0, 1,0) = (0,
1,0)."
=
(0,0,1)
= e3,
Tn (e3) = Tn (0, 0, 1) = (0,0,
1)."
=
(1,0,0)
=
ej,
whichgiverisetothe
matrix
(
00
1)
Tn = 1 0 0 .
o 1 0
The
reader
may
construct
the
other
five
matrices
of this
representation
andverify
directly
that it is
indeed
a
(faithful)
representation:
Products
and
inverses
of
permutations
are
mapped
onto
products
and
inverses
of the
corresponding
matrices.
Ill!II
The utility of a representation lies in our comfort with the structure
of
vec-
tor spaces. The climax
of
such comfort is the spectral decomposition theorems
of (normal) operators on vector spaces of finite (Chapter 4) aud infinite (Chap-
ter 16) dimensions. The operators
T
g
,
relevaut to our present discussion, are, in
general, neither normal nor simultaueously commuting. Therefore, the complete
diagonalizability of all T
g'S
is out
of
the question (unless the group happens to be
abeliau).
The best thing next to complete diagonalization is to see whether there are
common
invariant
subspaces of the vector space 1i: carrying the representation.
We already know how to construct (minimal) "invariaut" subsets of
1i:: these are
precisely the
orbits
of the action
of
the group G on 1i:.
The
linearity of T
g'S
gnarautees that the spau of each orbit is actually au invariaut subspace, aud that
such
subspaces
arethesmallest
invariant
subspaces
containing
agiven
vector.
OUf
aim is to find those
minimal
invariaut subspaces whose orthogonal complements
are also invariaut. We encountered the same situation in Chapter 4 for a single
operator.
reducible
and
irreducible
representations
24.1
DEFINITIONS
AND
EXAMPLES
677
24.1.7. Definition. A representationT : G
-+
GL(:Ji) is calledreducible ifthere
exist subspaces 11and W of:Ji such that :Ji = 11$
Wand
both 11and
Ware
invariant under all T
8s.
If
no such subspaces exist,:Ji is said to be irreducible.
In most cases of physical interest, where :Ji is a Hilbert space, W =
11.L.
Then, in the language
of
Definition 4.2.I, a representation is reducible if a proper
subspace
of:Ji
reduces all T
g's.
24.1.8.
Example.
Let83acton
~3
asinExample24.1.6.Forthecarrierspace%, choose
the space
of
functions on
IR
3,
and for
T,
the homomorphism T : G --+ GL(9-C), given by
Tgt(x)
=
t(x·
g), for t E K Anyt thatis symmetric inx, y, z,snchasxyz, x+y +z,
or
xl
+y2 +z2, defines a one-dimensional invariant subspace
of
Jf, To obtain another
invariant subspace, consider VI
(x, y, z) es
xy
and let {1fil?=l be as given in Example
23.4.1. Then, denoting
T1l'i
by Tj. the reader
may
check that
[Tltll(X, y, z) =
tl
«x,
y, z) .
"1)
=
tl
(x, y, z) =
xy
=
tl
(x, y, z),
[T2tll(X,
y, z) =
tl
«x,
y, z) .
"2)
=
tl
(y, x, z) =
yx
=
tl
(x, y, z),
[T3tll(X,
y, z) =
tl
«x,
y, z) .
"3)
=
tl
(z, y, x) = zy
==
t2(X,
y, z),
[T4tll(X,
y, z) =
tl
«x,
y, z) .
"4)
=
tl
(x, z, y) = xz
==
t3(X,
y, z),
[T5tll(X,
y, z) =
tl
«x,
y, z) .
"5)
=
tl
(z,x, y) = zx =
t3(x,
y, z),
[T6tll(x, y, z) =
tl
«x,
y, z) .
"6)
=
tl
(y, z, x) = yz =
t2(X,
y, z).
This is clearly a three-dimensional invariant subspace
of
9{ with
1/1'1,
1frz,
and
1fr3
as a
convenient basis,
in which the first three permutations are represented by
(
1 0 0) (1 0
0)
(0
1
0)
Tl = 0 1 0 ,
T2
= 0 0 1 ,
T3
= 1 0 0 .
001
010
001
It
is instructive for the reader to verify these relations and to find the three remaining
matrices.
II
24.1.9.
Example.
Let83acton
~3
as inExample24.1.6.Forthe catrierspaceofrepre-
sentation, choosethe subspace V
of
the
J(
of
Example 24.1.8 spanned bythe six functions
x, y, z,
xy,
xz,
and yz.
For
T, choose the
same
homomorphism as in Example 24.1.8 re-
strictedto V.
It
is clearthatthe subspaces
ti
andW spanned,respectively, bythefirstthree
and the last three functions are invariant under 83, and that V =11
EEl
W. It follows that
the representation is reducible.
The
matrix form
of
this representation is found to be
of
the
general form
(~g),
where B is one
of
the 6 matrices
of
Example 24.1.8. The matrix A,
correspondingto the threefunctions x, y, and z, can be found similarly. II
Let
Jf
be a carrierspace, finite- or infitrite-dimensional. For any vector
10),
the
reader may check that the span
of
(Tg
10)
}gEG
is an invariant subspace of :Ji.1f G
is finite, this subspace is clearly fitrite-dimensional. The irreducible subspace con-
taining
10),
a subspace of the span of
(T.
la)}gEG, will also be finite-dimensional.
Because of the arbitrariness
of
10),
it follows that every vector of :Ji lies in an
irreducible snbspace, and that
678 24.
GROUP
REPRESENTATION
THEORY
24.1.10. Box. All irreducible representations
of
a finite group are finite-
dimensional.
All
representations
are
equivalent
to
unitary
representations.
Due to the importance and convenience
of
unitary operators (forexample, the
fact that they leave the innerproductinvariant), it is desirableto be able to construct
a unitary representation--ora representation that is equivalent to one-s-ofgroups.
The following theorem ensures that this desire
can
be realized for finite groups.
24.1.11.
Theorem.
Every representation
of
afinitegroup G is equivalentto some
unitary representation.
Proof
We present the proof because
of
its simplicity and elegance. Let T be a
representation
of
G. Consider the positive hermitian operator T
==
LXEG
TtTx
and note that
Ting
=
~)T(g)]t[T(x)]tT(x)T(g)
xEG
=
I)T(xg)]tT(xg)
=
L[T(y)]tT(y)
=T,
xeG
yeG
(24.2)
v g
EG.
where we have used the fact that the sum over x and y
==
xg
sweep through the
entire group. Now let S
= ..(f, and multiply both sides of Equation
(24.2)-with
S2 replacing
T-by
S-I
on the left and by
T;-IS-I
on the right to obtain
S-ITt
S
-
ST-1S-
1
=}
(ST
S-I)t
- (ST
S-I)-I
g
-g
g - g
This shows that the representation
T'
defined by
T~
sa STgS-1 for all g
EGis
unitary. D
There is another convenience afforded by unitary representations:
24.1.12.
Theorem.
Let T : G ->- GL(:Ji) be a unitary representation and W an
invariant subspace of:Ji. Then,
W.l
is also invariant.
Proof
Suppose la) E
W.l.
We need to show that T
g
la) E
W.l
for all g E G. To
this end. let Ib) E W. Then
(blT
g
la) =
«al
Ti Ib))* =
«al
Tg1Ib))* =
«al
Tg-I Ib))* = 0,
because Tg-I Ib) E W. It follows from this equality that T
g
la) E
W.l
for all
gEG.
D
The carrier space :Jiof a unitary representation is either irreducible or has an
invariant subspace W, in whichcasewe have
:Ji= W
Ell
w-, where, by Theorem
24.1.12,
W.l
is also invariant.
If
W and
w-
are not irreducible, then they too can
(24.4)
antisymmetric
representation
ofa
permutation
group
adjoint
and
complex
conjugate
representations
24.1 DEFINITIONS
AND
EXAMPLES
679
be written as direct sums
of
invariant subspaces. Continuing this process, we can
decompose
J{
into irreducible invariant subspaces
W(k)
such that
J{
=
W(I)
Ell
W(2)
Ell
W(3)
Ell
....
If
the carrier space is finite-dimensional, which we assume from now on and for
which we use the notationV, then the above direct sum is finite and we write
p
V =
W(l)
Ell
W(2)
Ell
...
Ell
W(p)",
L
EllW(k).
(24.3)
k~1
One can think
of
W(k)
as the carrier space
of
an (irreducible) representation.
The
homomorphism T(k) : G --+ GL(W(k») is simply the restriction
of
T to the
subspace
W(k),
and we write
r
Ts = T1
1
)
Ell
T1
2
)
Ell
...
Ell
Tr)
sa L
Ell
T1
k
) •
k~1
If
we identify all equivalentirreducible representations and collect them together,
we
may
rewrite the last equation as
P
T = mlT(l)
ffi
m2T(2)
ffi
•••
ffi
m
T(P)
=
"Ellm
T(a)
g « > « >
WPg-L-
ag'
a=l
where p is the number of inequivalent irreducible representations and m
a
are
positive integers giving the number
of
times an irreducible representationT1
a
) and
all its eqnivalents occurin a given representation.
In
terms
of
matrices, Tg will be represented in a block-diagonal form as
',{f:'
jJ
g
where some
of
the
T1
k
)
may
be equivalent.
24.1.13. Example. A one-dimensional (and therefore irreducible) representation, de-
finedforallgroups,isthetrivial (symmetric) representation T : G
-+
iC
givenbyT (g) =t
for all g E G. For the permutationgroup
Sn,
one can define another one-dimensional (thus
irreducible) representation
T : Sn
~
C, called the antisymmetric representation, given
by
T(lf)
= +1
if
rr iseven,and
T(lf)
=
-1
if
If
is odd.
III
Given any (matrix) representation T
of
G, one
can
form the transpose inverse
matrices
(T~)-I,
and complex conjugate matrices
T;.
The reader
may
check that
each set
of
these matrices forms a representation
of
G.
24.1.14. Definition. The
set
of
matrices
(T~)
-1
and
T; are called,respectively, the
adjointrepresentation, denotedby
'1,
and
the
complex
conjugaierepresentation,
denoted by T*.
680 24.
GROUP
REPRESENTATION
THEORY
24.2 Orthogonality Properties
Homomorphisms preserve group structures. By studying a group that is more
attuned to concrete manipulations, we gain insight into the structure of groups
that are homomorphic to it. The group
of
invertible operators on a vector space,
especially in their matrix representation, are particularly suited for such a study
because of our familiarity with matrices and operators. The last section reduced
this study to inequivalent irreducible representations. This section is devoted to a
detailed study of such representations.
Schur's
lemma
24.2.1.
Lemma.
(Schur's lemma)
Let
T : G
--+
GL(V)
and
T'
: G
--+
GL(V')
be irreducible representations
ofG.
If
A E I:(V, V') is such that
v g E G,
(24.5)
v g
EG.
then either A is an isomorphism (i.e., T is equivalent to T'), or A = O.
Proof
Let la) E ker A.Then
ATgla)
=~Ala)
=0
=}
Tgla}
EkerA
'-,,-'
=0
It
follows that ker A, a subspace
of
V, is invariant under T. Irreducibility
of
T
implies that either ker A = V, or ker A =
O.
The first case asserts that Ais the zero
linear
transformation;
thesecondcase impliesthatAis
injective.
Similarly, let Ib) E A(V). Then Ib} = A[x) for some Ix) E V:
~ Ib} =
~A
Ix) = AT
g
Ix)
=}
T
g
Ib} E A(V)
.
'---,---'
EA(I')
v g E G.
It
follows that A(V), a subspace
of
V', is invariant under T'. Irreducibility of T'
impliesthateitherA(V) = O,orA(V) = V'. The firstcase is consistentwith thefirst
conclusion drawn above: ker A = V. The second case asserts that Ais surjective.
Combining the two results, we conclude that A is either the zero operator or an
isomorphism. D
Lemma 24.2.1 becomes extremely useful when we concentrate on a single
irreducible
representation,
i.e., whenT' = T.
24.2.2.
Lemma.
Let T : G
--+
GL (V) be an irreducible representation
of
G.
If
A E I:(V) is such that AT
g
=
TgAfor
all g E G, then A =
A1.
Proof
Replacing V' with V in Lemma 24.2.1, we conclude that A = 0 or A is
an isomorphism
of
V.
In
the first case, A =
O.
In the second case, Amust have a
nonzero eigenvalue Aand at least one eigenvector (see Theorem 4.3.4). It follows
that the operatorA- A1 commutes with
all Tg'Sand it is not an isomorphism (why
not?). Therefore, it must be the zero operator. D
24.2 ORTHOGONALITY PROPERTIES 681
We can immediately put this lemma to good use.
If
G is abelian, all operators
{TxlxeG commute with oue another. Focusing on one
of
these operators, say T
g
,
noting that it commutes with all operators
of
the representation, and usiug Lemma
24.2.2, we conclude that T
g = A1.
It
follows that wheu Tg acts ou a vector, it gives
a multiple
of
that vector.Therefore, it leaves any one-dimeusioual subspace of the
carrier space invariant. Since this is true for all
g E G, we have the following
result.
24.2.3.Theorem. All irreducible representations
of
an abelian group are one-
dimensional.
This theorem is an immediate consequeuce
of
Schur's lemma, and is independent
of the order of G.
In
particular, it holds for infinite groups,
if
Schur's lemma holds
for those groups. One importantclass of infinite groups for which Schur's lemma
holds is theLie groups.Thus,
allabelian
lie
groups have I-dimensionalirreducible
representations. We shall see later that the converse of Theorem 24.2.3 is also true
for finite groups.
Issai Schur (1875-1945)wasoneofthemostbrilliaat
math-
ematicians
active in
Germany
during the
first
third
of the
twentieth
century.
He
attended
the
Gymnasium
inLibau(now
Liepaja,
Latvia)
andthenthe
University
of
Berlin,
wherehe
spent
mostcfhis scientific
career
from
1911
until1916.When
he
returned
to
Berlin,
he was an
assistant
professor
at
Bonn.
He
became
full
professor
at
Berlin
in 1919.
Schur
was
forced
to
retire
by theNazi
authorities
in 1935butwas ableto em-
igrate
to
Palestine
in 1939.He died
there
of a
heart
ailment
several
years
later.
Schur
hadbeena
member
of the
Prussian
Academy
of SciencesbeforetheNazi
purges.
He
married
and
hadason and
daughter.
Schur's
principal
fieldwasthe
representation
theory
of
groups,
founded
alittlebefore
1900 by his
teacher
Frcbenlus.
Schur
seems to have
completed
this field
shortly
before
World
War
I, but he
returned
to the
subject
after 1925, whenit
became
important
for
physics.
Further
developed
by his
student
Richard
Brauer,
it is in ourtime
experiencing
an
extraordinary
growth
through
the
opening
of
new
questions.
Schur's
dissertation
(1901)
became
fundamental
tothe
representation
theory
ofthe
general
linear
group;
in
fact,
English
mathematicians
have
named
certain
of the
functions
appearing
in thework
"S-functions"
in
Schur's
honor.
In 1905
Schur
reestablished
the
theory
of
group
characters-the
keystone
of
representation
theory.
Themost
important
toolinvolved is
"Schur's
lemma."
Alongwith
the
representation
of
groups
by
integral
linear
substitutions,
Schur
was also the
first
to
study
representation
by
linear
fractional
substitutions,
treating
thismore
difficult
problem
ahnost
completely
in two
works
(1904,1907).In 1906Schur
considered
the
fundamental
problems
that
appear
whenan
algebraic
number
fieldis
taken
as the
domain;
a
number
appearing
in this
connection
is now calledthe
Schur
index.His
works
written
after
1925
include
a
complete
description
of the
rational
andof the
continuous
representations
of the
general linear
group;
the
foundations
of this
work
wereinhis
dissertation.
682
24.
GROUP
REPRESENTATION
THEORY
A lively interchange
with
many colleagues
led
Schur
to contribute important memoirs
to other areas
of
mathematics.
Some
of
these
were
published as collaborations with other
authors, although publications with
dual
authorship
were
almost unheard
of
at that time.
Herewesimplyindicatethe areas:pure grouptheory,matrices,algebraicequations,number
theory, divergent series, integral equations, and function theory.
24.2.4.
Example.
SupposethattheHamiltonian H
of
aquantummechanical systemwith
Hilbert space n has a group
of
symmetry with a representation T : G
~
GL(9-C).
Then
HTg = TgH for all g E G.
It
follows that H = A1 if the representation is irreducible.
Therefore,
24.2.5. Box.
All
vectors
of
each invariant irreducible subspace are eigenstates
of
the hamiltonian corresponding to the same eigenvalue, i.e., they all have the same
energy. Therefore, the degeneracy
of
that energy state is at least as large as the
dimension
of
the carrier space.
It
is helpful to arrive at the statement above
from
a different perspective. Consider a
vector
Ix} in the eigenspace JV(j correspondingto the energy eigenvalue Ei. Since Tg
and
H
commute, Tg [x} is also in JV(i. Therefore, an eigenspace
of
a Hamiltonian
with
a group
of
symmetry is invariant under all Tg for any representation T
of
that group. ITT is one
of
the
irreducible representations
of
G, say
T(a)
with dimension n
a,
then
dimMi
2:n
a.
II
Considertwo irreducible representations
TCa)
and T(P)
of
a group G with car-
rier spaces
w(a)
and W(fJ),respectively. Let Xbe any operator in
,C(WCa),
W(P»),
and define
A
ea
LT~a)XT~}I
= L TCa)(x)XTCP)(x-
1
) .
xeG
xeG
Then, we have
T~a)A
=
LT(a)(g)TCa)(x)XTCP)(x-1)T(fJ)(g-I)T(P)(g)
XEG
=
LT(a)(gx)XT(P)«gx)-l)
T(P)(g) = AT;fl.
xEG
=A because this sum also covers all G
We are interested in the two cases where T(a) = T(P), and where T(a) is not
equivalent to
T(P). In the first case, Lemma 24.2.2 gives A = A1;in the second
case, Lemma 24.2.1 gives A
= O.Combining these two results and labeling the
constant multiplying the unit operator by
X, we
can
write
LTCa)(g)XT(P)(g-l)
= AX
8ap1.
gEG
(24.6)