8.2 Representations ofSU(2): construction
The argument of the previous section only tells us what properties possible
finite dimensional irreducible representations ofSU(2) must have. It shows
how to construct such representations given a highest weight vector, but does
not provide any way to construct such highest weight vectors. We would like
to find some method to explicitly construct an irreducible (πn,Vn) for each
highest weightn. There are several possible constructions, but perhaps the
simplest one is the following, which gives a representation of highest weightn
by looking at polynomials in two complex variables, homogeneous of degreen.
This construction will produce representations not just ofSU(2), but of the
larger groupGL(2,C).
Recall from equation 1.3 that if one has an action of a group on a spaceM,
one can get a representation on functionsfonMby taking
(π(g)f)(x) =f(g−^1 ·x)For the groupGL(2,C), we have an obvious action onM=C^2 (by matrices
acting on column vectors), and we look at a specific class of functions on this
space, the polynomials. We can break up the infinite dimensional space of
polynomials onC^2 into finite dimensional subspaces as follows:
Definition(Homogeneous polynomials). The complex vector space of homo-
geneous polynomials of degreenin two complex variablesz 1 ,z 2 is the space of
functions onC^2 of the form
f(z 1 ,z 2 ) =a 0 zn 1 +a 1 z 1 n−^1 z 2 +···+an− 1 z 1 zn 2 −^1 +anzn 2The space of such functions is a complex vector space of dimensionn+ 1.
It turns out that this space of functions is exactly the representation space
Vnthat we need to get the irreducible representations ofSU(2)⊂GL(2,C). If
we take
g=(
α β
γ δ)
∈GL(2,C)
to act byg−^1 ·x=g−^1 xon column vectorsx∈C^2 , then its action on coordinates
z 1 ,z 2 will be given by
g·(
z 1
z 2)
=gT(
z 1
z 2)
=
(
α β
γ δ)T(
z 1
z 2)
=
(
αz 1 +γz 2
βz 1 +δz 2)
This is because these coordinates are basis elements of the dualC^2 , see the
discussion at the end of sections 4.1 and 4.2.
The representationπn(g) on homogeneous polynomial functions will be given
by replacing
z 1 →αz 1 +γz 2 , z 2 →βz 1 +δz 2
in the expression for the polynomial in terms ofz 1 andz 2.