representations (πn,Vn) ofSU(2), we know the characters to be the functions

χVn

((

eiθ 0

0 e−iθ

))

=einθ+ei(n−2)θ+···+e−i(n−2)θ+e−inθ (9.2)

Asngets large, this becomes an unwieldy expression, but one has

Theorem(Weyl character formula).

χVn

((

eiθ 0

0 e−iθ

))

=

ei(n+1)θ−e−i(n+1)θ

eiθ−e−iθ

=

sin((n+ 1)θ)

sin(θ)

Proof.One just needs to use the identity

(einθ+ei(n−2)θ+···+e−i(n−2)θ+e−inθ)(eiθ−e−iθ) =ei(n+1)θ−e−i(n+1)θ

and equation 9.2 for the character.

To get a proof of 9.1, compute the character of the tensor product on the di-
agonal matrices using the Weyl character formula for the second factor (ordering
things so thatn 2 > n 1 )

χVn (^1) ⊗Vn 2 =χVn 1 χVn 2
=(ein^1 θ+ei(n^1 −2)θ+···+e−i(n^1 −2)θ+e−in^1 θ)
ei(n^2 +1)θ−e−i(n^2 +1)θ
eiθ−e−iθ

(ei(n^1 +n^2 +1)θ−e−i(n^1 +n^2 +1)θ) +···+ (ei(n^2 −n^1 +1)θ−e−i(n^2 −n^1 +1)θ)
eiθ−e−iθ
=χVn 1 +n 2 +χVn 1 +n 2 −^2 +···+χVn 2 −n 1
So, when we decompose the tensor product of irreducibles into a direct sum of
irreducibles, the ones that must occur are exactly those of theorem 9.1.

9.4.3 Some examples

Some simple examples of how this works are:

• Tensor product of two spinors:

V^1 ⊗V^1 =V^2 ⊕V^0

This says that the four complex dimensional tensor product of two spinor

representations (which are each two complex dimensional) decomposes

into irreducibles as the sum of a three dimensional vector representation

and a one dimensional trivial (scalar) representation.

Using the basis

(

1

0

)

,

(

0

1

)

forV^1 , the tensor productV^1 ⊗V^1 has a basis

(

1

0

)

⊗

(

1

0

)

,

(

1

0

)

⊗

(

0

1

)

,

(

0

1

)

⊗

(

1

0

)

,

(

0

1

)

⊗

(

0

1

)