Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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232 Chapter
16.
Representations of Compact Simple
Lie
Algebras
This
can
and
must
be
inverted. Let us define
the
Canan
matrix A,
a
reall
x
l
matrix,
as
(16.25)
It
has extremely simple elements. Recalling
that
the
g(a)
are simple roots, we
have
the
values:
Aaa
=
2,
a#-
b:
Aab
=
0 for
g(a).
Q(b)
=
0,
i.e.,
0
0,
=
-1
for
0-0
=
-lor
- 2 for
0=0
=
-lor
- 3 for
O:=:O
(16.26)
Even
though
this
matrix
is
not symmetric,
the
simple values of its entries make
it
easy
to
work with.
If
we
wish, we
can
express A as
the
product
of
a symmetric
and
a diagonal matrix:
A=SD,
Sab
=
Sba
=
Q(a)
.
Q(b)
,
Dab
=
20
a
bl[g(a)
[2
(16.27)
Linear independence of
the
simple
roots
means
that
A
is nonsingular,
but
A-
l
is
again
not
symmetric.
On
the
other
hand,
the
matrix
(16.28)
is
symmetric. We shall express
the
solution for
the
highest weight
A
in
terms
of
the
set
of
integers
{N
a
}
by using
this
matrix
G.
The
equation
to
be
inverted,
Eq.(16.24), reads in
matrix
form:
(16.29)
so
the
solution is
(16.30)
Writing
out
the
individual components
and
putting
them
into Eq.(16.23),
2
I
.xa
=
IQ(a)
12
t;
GabNb
A
=
L
2G
a
b
N
bQ(a)
IIQ(a)1
2
(16.31 )
a,b
This
way
of
writing
A
in
terms
of
{N
a
}
leads
to
the
next
important
step.
We said
that
any
choice of
the
N's
leads
to
a unique
UIR
and
vice versa. We
can
then
consider
the
l
special
UIR's
obtained
by
setting
all
Na
but
one zero,
16.4.
Fundamental UIR's, Survey of all UIR's
233
and
taking
the
non-zero one
to
be
unity.
In
this way
certain
special highest
weights
must
arise. We
write
them
as
J(a)
=
L2G
ab
g(b)/lg(b)1
2
b
J
=
L
NaJ
(a)
a
We use
these
particular
highest weights
in
the
following section.
(16.32)
16.4
Fundamental
VIR's,
Survey
of
all
VIR's
Cartan
has
shown
that
for
any
CSLA,
there
are
I
basic
or
fundamental
UIR's
out
of
which all
others
can
be
built
via
symmetrised
direct
products.
We
denote
the
fundamental
UIR's
by
v(a),
a
=
1,2,
...
,
I:
v(a)
=
a
th
fundamental
UIR
with
highest weight
J(a)
=
UIR
with
Na
=
1,
Nb
=
0 for
b
=1=
a
(16.33)
In
this
sense, these
J
(a)
are
the
simplest highest weights:
they
are
called
fundamental dominant weights.
Any
dominant
weight (i.e.,
the
highest in
an
equivalence class
under
the
Weyl group) is a non-negative
integral
combination
of
the
J
t
a
).
The
highest weight
of
a
UIR
is
the
linear
combination
(16.32).
The
UIR
uniquely
determined
by
{N
a
}
can
be
indicated
by
use
of
the
Dynkin
diagram
for
the
group.
On
this
diagram
we
mark
the
positions
of
the
first, second,
...
,
[th
simple
roots
g(1),
g(2),
...
gO).
Then
just
below
or
at
the
corresponding circles we
write
the
values
of
N
l
, N
2
,
...
, N
1
•
In
this
depiction,
v(a)
has
unity
at
the
a
th
root
position, zero elsewhere.
How is
the
UIR
V
==
{N
a
}
obtained
from
the
fundamental
UIR's
v(a)?
The
prescription is
to
take
Nl
products
of
V(1),
N2
of
V(2),
...
,N
1
of
V(l),
take
the
direct
product
of
them
all,
and
isolate
the
highest weight in
this
product.
This
highest weight is evidently given by Eq.(16.32).
Thus
on
reduction
of
this
direct
product
representation,
the
"largest piece" is
the
UIR
{N
a
}
we
are
after:
V(1)
x
...
x
V(1)
x
V(2)
x
...
V(2)
x
...
DO)
x
...
V(l)
-->
+--
Nl
---">
(16.34)
The
fundamental
dominant
weights
J(a)
determining
the
fundamental
UIR's
out
of
which all
UIR's
can
be
built, have a simple geometrical relationship
to
the
simple
roots
S.
Since each
J(a)
corresponds
to
Na
=
1,
Nb
=
0 for
b
=1=
a,
we see
that
(16.35)
We therefore
set
up
the
following procedure.
Starting
with
the
simple
roots
g(a),
we rescale
them
to
define
the
linearly
independent
(but
not
normalised!)
234
Chapter
16.
Representations of Compact Simple
Lie
Algebras
vectors
(16.36)
Then
the
fundamental
dominant
weights form
the
reciprocal basis
in
root
space
to
this
set of vectors
.J(a) .
a
Jb
)
=
Jab
(16.37)
Incidentally, these vectors
a
Ja
)
are related
to
the
symmetric
matrix
0
ap-
pearing in Eqs.(16.28),(16.30),(16.31),(16.32):
0-
1
=
DSD
=
(oJa
l
.OJb
l
);
(16.38)
The
integers
{N
a
}
label
VIR's
uniquely,
and
they
tell us graphically how
to
build
up
general
VIR's
from
the
fundamental ones. Are
there
invariant
operators
such
that
the
Na
or
functions
thereof
are
eigenvalues of these operators?
There
are, namely
the
Casimir
operators
which
are
(symmetric) polynomials formed
out
of
the
generators
X
j
.
For example,
the
simplest
quadratic
one is
·k
{12
=
gJ
XjXk
I
=
I::
H~
+
I::
E-aEa
(16.39)
a=l
aE!R
Then
there
are cubic
and
higher order expressions, in fact precisely
Z
independent
Casimir operators. However we will
not
pursue
their
properties in any detail.
For a
CSLA
of
rank
l,
when is
the
VIR
{N
a
}
real
or
potentially real?
When
is
it essentially complex?
Just
as a
VIR
has a unique simple highest weight
.4,
it
also has a lowest simple weight
fl...
Then
the
VIR
{N
a
}
is
equivalent
to
its
complex conjugate if
and
only if
.4
=
-fl...
More generally, if
.4
and
fl..
are
the
highest
and
lowest weights of
the
VIR
V,
then
-fl..
and
-.4
are
the
highest
and
lowest weights of
the
VIR
V*.
From
this
general criterion we see immediately how it
is
that
each
VIR
of
SV(2)
is self conjugate:
the
maximum
and
minimum
values of
m,j
and
-j,
are
negatives of one another.
We have seen
that
the
complete list
of
compact
simple Lie groups
is
SV(n)
for
n
;:::
2;
SO(n)
for
n
;:::
7;
Vsp(2n)
for
n
;:::
2;
02;
F4;
En
for
n
=
6,7,8.
Which
of these possess complex VIR's? A detailed amilysis shows
that
only
SV(n)
for
n
;:::
3, SO(
4n
+
2)
for
n
;:::
2
and
E6
have some complex VIR's; in all
other
cases
each
VIR
is
real or pseudo real.
16.5
Fundamental
VIR's
for
A
l
,
B
l
,
G
l
,
D
z
The
simple
roots
for each of these groups have been given in
Chapter
14.
In
principle
we
can
then
calculate
the
fundamental
dominant
weights as
the
basis
reciprocal
to
{&(a
l
}.
The
calculations are quite easy for
Bl,
Cl
and
D
l
,
and
a
bit
involved for
AI.
We will
take
them
up
in
the
same sequence
Dl,
Bl,
Cl,
Al
in
which we dealt
with
them
in
Chapter
14.
16.5.
Fundamental UIR's
for
AI,
B
l
,
Cl,
Dl
235
Case
of
Dz
SO(21):
With
~
being
the
standard
unit
vectors in Euclidean i-dimensional space, we
have from Eq.(14.20)
the
following set of simple roots:
gel)
=
fl
-
f2'
g(2)
=
f2
-
fJ,
...
,
g(l-l)
=
fl-l
-
fl'
g(l)
=
fl-l
+
fl
(16.40)
Since each
Ig(a) I
=
y'2,
the
rescaling (16.36)
has
no effect:
g(a)
=
&(a)
(16.41)
So
the
fundamental
dominant
weights
01(1),01(2),
...
form a
system
recipro-
cal
to
the
set (16.40).
At
this
point
let us remember from Eq.(14.13)
that
in
the
defining vector
representation
D of SO(2i),
the
highest weight
is
fl.
In
fact,
the
set of all weights in
this
representation
can
be
arranged
in decreasing
order
thus:
WD
= weights in defining representation
D
of
SO(2i)
=
{fl,f2'···
,fl'
-fl'···
'
-f2
'
-fl}
(16.42)
We will use
this
information in
interpreting
the
fundamental representations.
Let
us now list
the
fundamental
dominant
weights
o1(a);
they
are easy
to
find from
the
condition
that
they
form a basis reciprocal
to
the
simple
roots
(16.40):
A(l)
=e
A(2)
=e
+e
-
-1'
-
-1
-2'
A
(3) -
_ -
fl
+
f2
+
f3'
...
,
o1Y-2)
=
fl
+
f2
+
...
+
fl-2,
A
(l-l)
_
1 ( )
- -
"2
fl
+
f2
+
...
+
fl-l
-
fl
,
(l)
1
L1
=
"2(fl
+
f2
+
...
+
fl-l
+
fl)
. (16.43)
The
"half integral"
structure
of
the
last two
must
come as somewhat of
a surprise!
What
can
we say
about
the
corresponding fundamental
VIR's
V(1),
V(2),
. . . ,
V(l)?
Comparing
with
the
weights
WD
of
the
defining repre-
sentation
Din
Eq.(16.42), we
can
easily see
that
V(1)
= defining
vector
representation D
V(2)
= representation on second
rank
antisymmetric
tensors,
(D
x D)antisymmetric
V(3)
= representation on
third
rank
antisymmetric
tensors,
(D
x
D
x D)antisymmetric
V(l-2)
=representation
on
(l -
2)
rank
antisymmetric
tensors.
(16.44)
236
Chapter
16.
Representations of Compact Simple
Lie
Algebras
The
truth
of
these
statements
would
be
evident
to
you
by
using
arguments
similar
to
those
in
quantum
mechanics when
enumerating
the
possible
states
of
identical fermions obeying
the
exclusion principle. So all
the
fundamental
UIR's
except the last two
really arise
out
of
the
defining
representation
D,
and
its
products
with
itself.
The
last
two, however,
can
never
be
obtained
by
any
finite algebraic
procedure
starting
with
D;
this
is evident
on
comparing
.!1Y-1)
and
.4(l)with
WD!
These
last
two "unexpected"
fundamental
UIR's
of
SO(21)
are
called
spinor representations.
V(l-l)
=
first spinor representation, dimension 2
Z
-
1
,
also
written
L1
(1);
v(l)
=
second spinor representation, dimension 2
Z
-
1
,
also
written
L1
(2)
(16.45)
We shall
study
the
construction
and
properties
of
these spinor
UIR's
in
the
next
Chapter.
At
this
point,
it
suffices
to
repeat
that
the
l
fundamental
UIR's
of
SO(21)
are
the
antisymmetric
tensor
representations
of
ranks
1,2,
...
,I
-
2,
and
the
two (inequivalent) spinor
representations
L1
(1),
L1
(2).
Associated
with
the
group
SO(2l) is
an
antisymmetric
Levi-Civita symbol
carrying
2l indices. So one knows
that
among
antisymmetric
tensors
there
is
no need
to
go
beyond
those of
rank
l.
One
naturally
asks: where
are
the
anti
symmetric
tensors
of
ranks
I-I
and
l,
and
what
status
do
they
have?
They
are
of course
not
in
the
list
of
fundamental
UIR's.
It
turns
out
that
the
(l-l)th
rank
antisymmetric
tensors furnish a
UIR
with
the
highest weight
.4
=
f.z
+
f.2
+ ... +
f.Z-1
=
.4(l-1)
+
.4(Z)
(16.46)
Thus,
in
the
{N
a
}
notation,
this is
the
UIR
(0,0,
...
,1,1),
and
it
occurs
in
the
direct
product
L1(1)
x
L1(2)
of
the
two spinor UIR's. As for
the
lth
rank
antisymmetric
tensors,
they
are
reducible into self
dual
and
antiself
dual
types,
and
depending
on
one's definitions, one is present
in
{L1(l) x
L1(1)}
symm
and
the
other
in
{L1
(2) X
L1
(2)
}symm.
We shall deal
with
these in some detail in
the
next
Chapter.
Case
of
B
z
=
SO(2l
+
1):
The
set
of
simple
roots
in
this
case is, from Eq.(14.28),
(1)
= _
(2) _ _
(Z-l)
_
(l)
Q
f.1
f.2'
Q -
f.2
f.3'·
..
,
Q -
f.z-z - f.z,
Q
=
f.z
(16.4
7)
After rescaling according
to
Eq.(16.36), we
get
the
vectors
dJa):
&(1)
= _
'(2)
_ _
,(Z-l)
_
(Z)
)
f.1
f.2'
a -
f.2
~,
...
,
a -
f.Z-1 - f.z,
&
=
2f.z
(16.48
16.5.
Fundamental UIR's
for
AI,
B
l
,
C
l
,
Dl
237
Now
the
set of weights in
the
defining representation,
arranged
from
the
highest
down
to
the
lowest, are:
WD
=
weights in defining representation
D
of
SO(2l
+
1)
=
{~1>~2""
'~I,Q,
-~l"'"
-~2'
-~d
(16.49)
We now have
the
requisite information
to
construct
the
fundamental dom-
inant
weights
and
interpret
the
corresponding fundamental UIR's:
the
former
are
A
(3) -
- -
~l
+
~2
+
~3'
...
,
We
are
thus
able
to
recognise
the
first
(l -
1)
fundamental UIR's:
v(1)
=
defining vector
representation
D
V(2)
=
representation
on
second
rank
antisymmetric
tensors,
(D
x
D)anti
s
ymmetric
V(l-l)
=
representation on
(l -
l)th
rank
antisymmetric
tensors
(16.50)
(16.51 )
In
contrast
to
SO(2l),
at
the
end of
this
list
there
is now only one spinor
UIR:
v(l)
=
unique spinor representation, dimension
2
1
,
written
.1
(16.52)
This
spinor representation can, of course, not
be
obtained
from
D
by any finite
algebraic procedure. We
study
it in detail in
Chapter
17.
For
SO(2l
+
1),
the
independent antisymmetric tensors are those of ranks
1,2,
...
,
l-l,
l.
All
but
the
last have
appeared
in
the
list of fundamental UIR's.
The
rank
l
antisymmetric tensors do
not
yield a fundamental UIR.
The
highest
weight for
their
representation
is clearly
the
sum
of
the
first
l
weights in
D
listed
in Eq.(16.49):
(16.53)
Therefore, in
the
{Na}
notation
they
furnish
the
UIR
(0,0,
...
,0,2),
and
this
is
the
leading
UIR
in
the
symmetric
part
of
the
product
.1
x
.1.
238
Chapter
16.
Representations of Compact Simple
Lie
Algebras
Case
of
C
z
=
USp(2l)
Interestingly,
the
simple
roots
and
the
scaled simple
roots
now
are
just
inter-
changed as
compared
to
BI
=SO(21
+
1): from Eq.(14.48),
g(1)
=
~1
-
~2'
g(2)
=~2
-~,
...
,
g(l-1)
=
~1-1
-
~I'
g(l)
=
2e'
-I'
OJ1) =
~1
-
~2'
OJ2)
=~2
-~,
...
,
&(1-1)
=
~1-1
-
~I'
&(1)
=~I
(16.54)
The
weights
of
the
defining
representation
given
in
Eq.(14.43)
are
the
same
as
with
SO(2l):
(16.55)
The
reciprocal basis
to
ala)
is very easily found.
There
are
no
half
integers
anywhere now:
.J(1)
=
~1'
.J(2)
=
~1
+
~2
+ ... ,
.J(1-1)
=
~1
+
~2
+ ... +
~1-1'
.J
(I)
=
~1
+
~2
+ ... +
f.1
(16.56)
There
is a uniform
pattern
now all
the
way
up
to
the
last
one. We
can
say
that
the
fundamental
VIR's
v(a),
a
=
1,2,
...
,
I,
are
given
by
the
ath
rank
antisymmetric
tensors over
the
defining vector
VIR
D,
except
that
in
this
sym-
plectic case
it
is
the
antisymmetric
objects
that
must
have
traces
removed!
The
trace
operation
of
course involves
contraction
with
the
symplectic
metric
'r/AB
of
Eq.(14.30). So
the
fundamental
VIR's
for VSp(21)
are
a
=
1,2,
...
,I:
V
Ca
)
=
a
th
rank
antisymmetric
traceless tensors
over defining vector
representation
D (16.57)
There
are
no spinors for
the
symplectic groups.
Case
of
Az
=
SU(l
+
1)
We have noticed in
Chapter
14
that
the
calculations
of
weights
and
roots
for
the
groups
SV(1
+
1)
tend
to
be
slightly messy
compared
to
the
other
families!
The
simple
roots
are
listed
in
Eq.(14.65).
Each
of
them
has
length
V2,
so we
have no change
on
rescaling. We
write
them
a
bit
more
compactly
as
(a)
_
A
(a)
_
1
(C
~
) _
I
g -
a -
Va+1
v
a~a
-
va
+
2~a+1
,a
-
1,2,
...
, - 1,
1-1
f¥
(I) _
A
(I)
_
'"
~a
I
+
1
a
~a
-
0
+
--el
-
a=l
Ja(a
+
1)
l-
(16.58)
Next, in listing
the
weights
of
the
defining
representation
D,
given in
Eq.(14.56),
it
is helpful
to
define
the
following unnormalised multiples
of the
unit
vectors
~a:
L
=~a/Ja(a+1),
a=1,2,
...
,1
(16.59)
16.5.
Fundamental
VIR's
for
AI,
E
l
,
G
l
,
Dl
239
Then
the
highest weight
in
D,
followed by
the
successively lower weights are:
p,(1)
= 1
+
1
+ ... +
1 ;
-
-1
-2
-I
!!(l+1)
=
-lL;
p,(I)
=
-(l
-
1)1
+
1 ;
-
-1-1
-I
!!:.(I-l)
=
-(l
-
2)L_2
+
L-1
+
L;
p,(3)
=
-21
+
1
+ ... +
1
+
f ;
-
-2 -3
-1-1-1
p,(2)
= -
f
+ 1 + ... + 1 +
f .
-
-1
-2
-1-1-1
(16.60)
Now
we
must
construct
the
1
fundamental
dominant
weights.
First
we recognise
the
role
of
the
defining
UIR
D.
Its
highest weight
!!:.(1)
is seen
to
obey
!!:.(1) .
oJa)
=
0,
a
=
1,2,
...
,l
-
1;
!!:.(1) .
oJI)
=
1
(16.61 )
Thus
according
to
the
general definition,
this
is
the
lth
fundamental
UIR:
!!:.(1)
= A(I),
D
==
V(/)
(16.62)
To find
the
other
fundamental
dominant
weights,
it
is helpful
to
expand
a
general vector in
the
form
I I
;r=
"£xaL
=
,,£Xa~a/Ja(a+1)
a=1 a=1
so
that
products
with
the
simple
roots
simplify:
;r'
oJa)
=
(xa
-
xa+d/(a
+
1),
a
=
1,2,
...
,
1 -
1;
1-1
(16.63)
;r.&(l)
=
,,£x
a
/a(a+1)+
~I.
(16.64)
a=l
Using such expansions, one
can
quickly discover
the
reciprocal basis
to
the
&(a):
A(a)
=
It
+
[2
+ ... +
L -
a(L+l
+ ...
+,0
a
I
=
,,£~/Jb(b
+
1)
-
a
"£
~b/Jb(b
+
1),
a
=
1,2,
...
,l-l
(16.65)
b=l
b=a+l
The
lth
fundamental
domin~nt
weight
A(l)
=
!!:.(1)
can
also
be
taken
to
be
given
by
this
same
general expression.
These
A(a)
can
now
be
expressed
in
terms
of
240
Chapter
16.
Representations of Compact Simple
Li
e Algebras
the
successive weights of
D
listed in Eq.(16.60), namely, beginning
with
(16.62):
J(/)
=
!:!:(1);
J(I-1)
=
!!.(1)
+
!!.(l+1)
;
J(I-2)
=
!!.(1)
+
!!.(l+1)
+
!!.(I);
(16.66)
These
expressions show
that
the
I fundamental
UIR's
of
Al
= SU
(l
+
1) are
the
defining representation
D ,
and
the
antisymmetric tensors over
D
of
ranks
2,
3,
...
,I
as one might have expected:
a
=
1,2,
...
:
Veal
=antisymmetric
tensors of
rank
(I
+
1 -
a)
over
D
(16.67)
As
with
the
symplectic groups,
there
are no spinors for
the
unitary
groups.
One might
be
tempted
to
rework
the
algebra for
the
Al
family of groups
to
avoid
the
rather
"peculiar"
patterns
for
J(a),
weights M in
D,
etc.
that
we
have found. However,
that
is
bound
to
introduce complications elsewhere, for
instance in
the
choice of
the
Cartan
sub
algebra
generators
Ha.
We have
made
what
seems like a simple
and
uncomplicated choice of
Ha
to
begin with,
and
then
gone
through
the
general
theory
in a systematic way.
16.6
The
Elementary VIR's
The
I
fundamental
UIR's
of a
rank
I
group
G
are
the
building blocks for all
UIR's
in
the
sense
that
the
general
UIR
{N
a
}
is
the
"largest" piece in
the
direct
product
of
N1
factors
V(1) , N2
factors
V(2),
....
It
is
isolated by working
with
the
highest weight in
this
product,
(16.68)
evidently lying in
the
product
of
the
completely symmetric
parts
of
{®V(1)}N
1
,
{®V(2)
}N2,
..
..
But
we have seen
that
for
AI,
B
I
,
GI
and
D
I
,
many
of
the
fundamental
UIR's
are
antisymmetric tensors over
the
defining
UIR
D!
Thus, if we have
the
defining
UIR
in
hand,
and
may
be
one or two more,
then
by
the
combined processes of
antisymmetrization
and
symmetrization of products, all
the
other
basic
UIR's
and
then
all
UIR
's
can
be
formed.
The
number
of elementary
UIR
's needed for
this
construction
is
just
2 or 3.
The
precise definition
is
that
they
correspond
to
the
"ends" of
the
Dynkin diagram.
So
in
the
sequence
D
I
,
B
I
, G
I
,
AI,
we have
16.7.
Structure of States within a UIR
241
the
elementary VIR's:
Dl
=SO(2l)
-
Three
elementary VIR's:
{N
a
}
=
{I,
O,
...
}
----+
vector VIR,
D;
{N
a
}
=
{O,
0,
...
,1,
O}
----+
spinor
L1(1)
{N
a
}
=
{O,
...
, 0,
I}
----+
spinor
L1(2)
Bl
=SO(2l
+
1)
- Two elementary VIR's:
{N
a
}
=
{I,
0,
...
,
O}
----+
vector VIR,
D;
{N
a
}
=
{O,O,
...
,0,
I}
----+
spinor
L1
Cl
=VSp(2l)
-
Two elementary
VIR's:
{Na}
=
{I,
0,
...
,
O}
----+
defining
VIR
D;
{N
a
}
=
{O,
...
,0,1}
----+
lth
rank
antisymmetric "traceless" tensors over
D.
Al
=SV(l
+
1) - Two elementary VIR's:
{N
a
}
=
{0,0,
...
,0,1}
----+
defining "vector"
VIR
D;
{N
a
}
=
{I,
0,
...
,O}
----+
lth
rank
antisymmetric tensors over
D.
However, in
the
C
l
and
Al
cases,
it
makes good sense
to
say
that
there
is
only one elementary UIR,
and
that
is
the
defining
VIR
D!
16.7
Structure
of
States
within
a
UIR
We have seen how
the
VIR's
of a
compact
simple Lie algebra
can
as a family be
classified,
and
also how
they
can
be
built
up
starting
from either fundamental
or elementary VIR's. We worked
out
all
the
relevant details for
the
classical
nonexceptional families of groups. Now let us briefly look
"inside" a generic
VIR
D,
to
get a feeling for
the
kinds of problems
and
structures
involved.
Given a
CSLA
I:
of
rank
l
and
order
r;
what
is
the
dimension of
the
VIR
{N
a
}?
Suffice
it
to
say
that
there
do exist explicit formulas
due
to
Weyl,
both
for
the
nonexceptional
and
the
exceptional cases.
The
former expressions are
given in Salam's lectures,
the
latter
are included in
the
book
by
Wybourne,
besides
many
other
places.
Let
W
be
the
set of all weights occurring in
D,.£1
the
highest weight,
and
/-L
a general one. While.£1
is
simple, in general
/-L
is
not:
this
is
the
multiplicity
problem.
In
the
most general
VIR
{N
a
},
it
turns
out
that
a basis vector in
V needs
~(r
-
3l)
additi~nal
independent
state
labels or
quantum
numbers
to
supplement
the
l weight vector components
/-La
which
are
the
eigenvalues
of
the
diagonal generators H
a
,
a
=
1,2,
...
, l.
Thus
, in general, a complete
commuting
set
within a
VIR
consists of
~(r
-
I) operators, l of
them
being
generators,
and
the
remainder
(in principle) functions of generators
(but
of
course
not
Casimir operators).
In
some
particular
VIR's,
it
could
happen
that