Proof.One way to prove this result is to use highest weight theory, raising
and lowering operators, and the formula for the Casimir operator. We will not
try and show the details of how this works out, but in the next section give a
simpler argument using characters. However, in outline (for more details, see
for instance section 5.2 of [71]), here’s how one could proceed:
One starts by noting that ifvn 1 ∈Vn 1 ,vn 2 ∈Vn 2 are highest weight vectors
for the two representations,vn 1 ⊗vn 2 will be a highest weight vector in the tensor
product representation (i.e., annihilated byπ′n 1 +n 2 (S+)), of weightn 1 +n 2.
So (πn 1 +n 2 ,Vn^1 +n^2 ) will occur in the decomposition. Applyingπn′ 1 +n 2 (S−) to
vn 1 ⊗vn 2 one gets a basis of the rest of the vectors in (πn 1 +n 2 ,Vn^1 +n^2 ). However,
at weightn 1 +n 2 −2 one can find another kind of vector, a highest weight vector
orthogonal to the vectors in (πn 1 +n 2 ,Vn^1 +n^2 ). Applying the lowering operator
to this gives (πn 1 +n 2 − 2 ,Vn^1 +n^2 −^2 ). As before, at weightn 1 +n 2 −4 one finds
another, orthogonal highest weight vector, and gets another representation, with
this process only terminating at weight|n 1 −n 2 |.
9.4.2 Characters of representations
A standard tool for dealing with representations is that of associating to a repre-
sentation an invariant called its character. This will be a conjugation invariant
function on the group that only depends on the equivalence class of the repre-
sentation. Given two representations constructed in very different ways, it is
often possible to check whether they are isomorphic by seeing if their character
functions match. The problem of identifying the possible irreducible represen-
tations of a group can be attacked by analyzing the possible character functions
of irreducible representations. We will not try and enter into the general theory
of characters here, but will just see what the characters of irreducible repre-
sentations are for the case ofG=SU(2). These can be used to give a simple
argument for the Clebsch-Gordan decomposition of the tensor product ofSU(2)
representations. For this we don’t need general theorems about the relations
of characters and representations, but can directly check that the irreducible
representations ofSU(2) correspond to distinct character functions which are
easily evaluated.
Definition(Character).The character of a representation(π,V)of a groupG
is the function onGgiven by
χV(g) =Tr(π(g))Since the trace of a matrix is invariant under conjugation,χV will be a
complex-valued, conjugation invariant function onG. One can easily check that
it will satisfy the relations
χV⊕W=χV+χW, χV⊗W=χVχWFor the case ofG=SU(2), any element can be conjugated to be in the
U(1) subgroup of diagonal matrices. Knowing the weights of the irreducible