This is sometimes called the “vector representation”, and we saw in chap-
ter 6 that it is isomorphic to the adjoint representation.
Composing the homomorphisms Φ andρ:π 2 =ρ◦Φ :SU(2)→SO(3)⊂GL(3,R)gives a representation (π 2 ,R^3 ) ofSU(2), the adjoint representation. Com-
plexifying gives a representation onC^3 , which in this case is just the action
withSO(3) matrices on complex column vectors, replacing the real coor-
dinates of vectors by complex coordinates.8.1 Representations ofSU(2): classification
8.1.1 Weight decomposition
If we make a choice of aU(1)⊂SU(2), then given any representation (π,V) of
SU(2) of dimensionm, we get a representation (π|U(1),V) ofU(1) by restriction
to theU(1) subgroup. Since we know the classification of irreducibles ofU(1),
we know that
(π|U(1),V) =Cq 1 ⊕Cq 2 ⊕···⊕Cqm
for someq 1 ,q 2 ,···,qm∈Z, whereCqdenotes the one dimensional representa-
tion ofU(1) corresponding to the integerq(theorem 2.3). Theseqjare called the
“weights” of the representationV. They are exactly the same thing discussed
in chapter 2 as “charges”, but here we’ll favor the mathematician’s terminology
since theU(1) here occurs in a context far removed from that of electromag-
netism and its electric charges.
Since our standard choice of coordinates (the Pauli matrices) picks out the
z-direction and diagonalizes the action of theU(1) subgroup corresponding to
rotation about this axis, this is theU(1) subgroup we will choose to define the
weights of theSU(2) representationV. This is the subgroup of elements of
SU(2) of the form (
eiθ 0
0 e−iθ
)
Our decomposition of anSU(2) representation (π,V) into irreducible represen-
tations of thisU(1) subgroup equivalently means that we can choose a basis of
Vso that
π(
eiθ 0
0 e−iθ)
=
eiθq^10 ··· 0
0 eiθq^2 ··· 0
··· ···
0 0 ··· eiθqm
An important property of the set of integersqjis the following:Theorem.Ifqis in the set{qj}, so is−q.