4Representations of Lie algebras

III Symmetries, Fields and Particles

4.5 Decomposition of tensor product of su

(2)

representa-

tions

We now try to find an explicit description of the decomposition of tensor products

of irreps of su(2).

We let

ρ

Λ

and

ρ

Λ

0

be irreps of

su

(2), where Λ

,

Λ

0

∈ N

. We call the represen-

tation spaces V

Λ

and V

Λ

0

.

We can form the tensor product ρ

Λ

⊗ ρ

Λ

0

with representation space

V

Λ

⊗ V

Λ

0

= span

R

{v ⊗ v

0

: v ∈ V

Λ

, v

0

∈ V

Λ

0

}.

By definition, for X ∈ su(2), the representation is given by

(ρ

Λ

⊗ ρ

Λ

0

)(X)(v ⊗ v

0

) = (ρ

Λ

(X)v) ⊗ v

0

+ v ⊗ (ρ

Λ

0

(X)v

0

).

This gives us a completely reducible representation of su(2) of dimension

dim(ρ

Λ

⊗ ρ

Λ

0

) = (Λ + 1)(Λ

0

+ 1).

We can then write

ρ

Λ

⊗ ρ

Λ

0

=

M

Λ

00

∈Z,Λ

00

≥0

L

Λ

00

Λ,Λ

0

ρ

Λ

00

,

where

L

Λ

00

Λ,Λ

0

are some non-negative integers we want to find out. These coefficients

are usually known as Littlewood-Richardson coefficients in general.

Recall that V

Λ

has a basis {v

λ

}, where

λ ∈ S

Λ

= {−Λ, Λ − 2, ··· , +Λ}.

Similarly, V

Λ

0

has a basis {v

0

λ

0

}.

Then we know that the tensor product space has basis

B = {v

λ

⊗ v

0

λ

0

: λ ∈ S

Λ

, λ

0

∈ S

Λ

0

}.

We now see what H does to our basis vectors. We have

(ρ

Λ

⊗ ρ

Λ

0

)(H)(v

λ

⊗ v

0

λ

0

) = (ρ

Λ

(H)v

λ

) ⊗ v

0

λ

0

+ v

λ

⊗ (ρ

Λ

0

(H)v

0

λ

0

)

= (λ + λ

0

)(v

λ

⊗ v

0

λ

0

).

We thus see that the weights of the tensor product are just the sum of the

weights of the individual components. In other words, we have

S

Λ,Λ

0

= {λ + λ

0

: λ ∈ S

Λ

, λ

0

∈ S

Λ

0

}

Note that here we count the weights with multiplicity, so that each weight can

appear multiple times.

We see that the highest weight is just the sum of the largest weights of the

irreps, and this appears with multiplicity 1. Thus we know

L

Λ+Λ

0

Λ,Λ

0

= 1,

i.e. we have one copy of

ρ

Λ+Λ

0

in the decomposition of the tensor product. We

write

ρ

Λ

⊗ ρ

Λ

0

= ρ

Λ+Λ

0

⊕ ˜ρ

Λ,Λ

0

,

where ˜ρ

Λ,Λ

0

has weight set

˜

S

Λ,Λ

0

satisfying

S

Λ,Λ

0

= S

Λ+Λ

0

∪

˜

S

Λ,Λ

0

.

We now notice that there is only one Λ + Λ

0

−

2 term in

˜

S

Λ,Λ

0

. So there must be

a copy of ρ

Λ+Λ

0

−2

as well. We keep on going.

Example. Take Λ = Λ

0

= 1. Then we have

S

1

= {−1, +1}.

So we have

S

1,1

= {−2, 0, 0, 2}.

We see that the highest weight is 2, and this corresponds to a factor of

ρ

2

. In

doing so, we write

S

1,1

= {−2, 0, +2} ∪ {0} = S

2

∪ S

0

.

So we have

ρ

1

⊗ ρ

1

= ρ

2

⊕ ρ

0

.

From the above, one can see (after some thought) that in general, we have

Proposition.

ρ

M

⊗ ρ

N

= ρ

|N−M|

⊕ ρ

|N−M|+2

⊕ ··· ⊕ ρ

N+M

.