14. Products of representations
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14. Products of representations
It has been shown that the infinitesimal generators {Mi } determine the
changes in vectors or operators induced by a group of transformations, by way
of Mi φ or [Mi , O]. In particular, if some vector or operator is left entirely
unchanged by any of the transformations of the group, i.e. it is invariant
under the action of the group, then it satisfies Mi φ = 0 or [Mi , O] = 0 for all
infinitesimal generators Mi . It is a basis for the trivial one-dimensional irrep
with Dynkin coefficients (0, 0, . . . , 0), called a singlet or scalar irrep.
As in the case of finite groups, larger representations can be formed by
multiplying irreps. Under this multiplication the basis can take the form of
the product of two vectors, φ(1) φ(2) , of an operator and a vector, Oφ, or of two
operators, O(1) O(2) . The representation so produced is generally reducible,
and one of the interesting questions is which irreps it contains. This can be
determined straightforwardly by the appropriate use of the known Dynkin
coefficients of the weights of the irreps.
Since, schematically, φ(1) φ(2) → Rφ(1) Rφ(2) → φ(1) φ(2) + (ǫMφ(1) )φ(2) +
φ (ǫMφ(2) ); while Oφ → ROR−1 Rφ → Oφ + [ǫM, O]φ + O(ǫMφ); and
O(1) O(2) → RO(1) O(2) R−1 → O(1) O(2) + [ǫM, O(1) ]O(2) + O(1) [ǫM, O(2)]; it is
clear that the weights associated with the product representation are simply
all the pairwise sums of weights of the irreps being multiplied, with their corresponding multiplicities. The level of a weight in the product representation
is the sum of the levels of weights added to produce it. In particular, the
highest weight of the product representation is just the sum of the highest
weights of the irreps being multiplied, with multiplicity one, and is at level
0.
(1)
This highest weight identifies the highest irrep contained in the product.
It, together with all the weights associated with it in the irrep, can be removed from the list of weights of the product irrep. The highest weight then
remaining (the one at the lowest level remaining), with its multiplicity, identifies the next highest irrep contained in the product representation, with its
multiplicity. Upon removal of all the weights of this irrep, with their corresponding multiplicities, the highest weight then remaining identifies the next
highest irrep in the product representation. The process continues until all
the weights of the product representation have been accounted for.
[As an illustrative example, consider the product of the (1, 0)
irrep of A2 with itself. This irrep contains three weights, of multiplicity one, with Dynkin coefficients (1, 0), (−1, 1), (0, −1), in
Introductory Algebra for Physicists
Michael W. Kirson
14. Products of representations
2
order of decreasing weight. The product (1, 0) ⊗ (1, 0) therefore contains nine weights — the highest weight (2, 0), once; the
next highest weight (0, 1), twice; the next highest weights (1, −1),
twice, and (−2, 2), once; the next highest weight (−1, 0), twice;
and the lowest weight (0, −2), once. The highest irrep contained
here is that associated with the highest weight, (2, 0). This 6dimensional irrep contains the weights (2, 0), (0, 1), (−2, 2), (1, −1),
(−1, 0), (0, −2), each once. Upon removing the (2, 0) irrep, there
remain the weights (0, 1), (1, −1), (−1, 0), each once, in order of
decreasing weight. These are precisely the weights of the (0, 1)
irrep. So, finally, (1, 0) ⊗ (1, 0) = (2, 0) ⊕ (0, 1).]
For any algebra, the product of any irrep with the singlet irrep is evidently
just the original irrep: Λ ⊗ (0, 0, . . . , 0) = Λ. It may also be noted that the
generators of the algebra, acting on the carrier space of an irrep, leave it
unchanged. (By definition, the carrier space is invariant under the action
of the generators.) But the generators constitute the adjoint representation,
denoted Λad , so their action on the irrep Λ is a concrete realisation of the
product Λad ⊗ Λ and produces Λ. But the reduction of Λad ⊗ Λ to irreps,
by the procedure outlined above, will generally give rise to a sum of irreps,
since the set of generators is not the only realisation of the irrep Λad . For
consistency, the general product Λad ⊗ Λ must contain Λ.
[For example, the adjoint representation of A2 is (1,1). According
to the argument just outlined, (λ, µ) ⊗ (1, 1) must contain (λ, µ),
for any irrep (λ, µ).]
Given the product of two irreps, labeled by their Dynkin indices, the
highest weight irrep can be read off by just adding the Dynkin indices of
the two factors, i.e. Λ(1) ⊗ Λ(2) contains the irrep Λ(1) + Λ(2) . Another irrep
contained in the product can be straightforwardly identified via Dynkin’s
second highest weight theorem. The method is based on an extension of the
Dynkin diagram of the algebra considered.
• Construct an extended Dynkin diagram by appending to the usual
Dynkin diagram two small circles representing the weights Λ(1) and
Λ(2) . Recall that each highest weight Λ(i) is a list of ℓ Dynkin in(i)
(i)
(i)
dices (λ1 , λ2 , . . . , λℓ ), all of which are non-negative integers, while
the usual Dynkin diagram shows all the simple roots of the algebra.
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Michael W. Kirson
14. Products of representations
3
Λ(1)
Λ(2)
e
e
1e 2e 3e . . . . . . . . . . . . ℓe
Dynkin diagram
-
Here the box represents the usual Dynkin diagram and the little circles
it contains represent the simple roots.
• Two elements β, γ of root space are said to be connected if β · γ 6= 0.
(i)
But the Dynkin index λj = 2Λ(i) · α(j) /α(j) · α(j) , so Λ(i) is connected
(i)
to the simple root α(j) if and only if λj > 0. To the extended Dynkin
diagram there are now added lines joining Λ(1) and Λ(2) to all the simple
(i)
roots to which they are connected, i.e. one line for each positive λj .
Λ(1)
Λ(2)
e
e
P
A
PPP
P
P
PP A
PA
P
1e 2e 3e . . . . . . . . . . . . ℓe
Dynkin diagram
-
• There is a chain of simple roots connecting Λ(1) and Λ(2) if there is a
sequence of connected simple roots α(i) → α(i+1) → α(i+2) → · · · → α(j)
whose ends are connected to Λ(1) and Λ(2) respectively and none of
whose internal roots are connected to Λ(1) or Λ(2) . Such a chain defines
a minimal connected path from Λ(1) to Λ(2) through the simple roots.
• If such a chain exists, then Λ(1) + Λ(2) − α(i) − α(i+1) − · · · − α(j) is the
highest weight of an irrep contained in Λ(1) ⊗ Λ(2) .
There is a consistency check on the reduction of the product of two irreps
to irreps, arising from the availability of the Weyl dimension formula. If
Λ(1) and Λ(2) have dimensions D1 and D2 respectively, and if the irreps Λi
in the reduction Λ(1) ⊗ Λ(2) = Λ1 ⊕ Λ2 ⊕ · · · ⊕ Λn have dimensions di , then
P
D1 D2 = ni=1 di . This can help determine when the reduction has been
completed.
Introductory Algebra for Physicists
Michael W. Kirson
14. Products of representations
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[Analogously to the case of finite groups, the reduction of a product of irreps to a direct sum of irreps produces a Clebsch-Gordan
series, with its associated Clebsch-Gordan coefficients. The WignerEckart theorem can be shown to hold for Lie groups and algebras
as well.]
Examples
1. The irreps of A1 are specified by the Dynkin coefficient (λ), where λ is a
non-negative integer and the irrep contains the λ+1 weights (λ), (λ−2), (λ−
4), . . . , (−λ), once each. It is then easy to see that the product (λ1 ) ⊗ (λ2 )
contains the weights (λ1 + λ2 ), once; (λ1 + λ2 − 2), twice; (λ1 + λ2 − 4), three
times; · · · (λ1 + λ2 − 2k), k + 1 times; · · ·. The highest irrep is (λ1 + λ2 ),
occurring once. When it is removed, the highest irrep remaining is (λ1 +
λ2 − 2), occurring once. When this is removed, the highest irrep remaining
is (λ1 + λ2 − 4), occurring once. This continues until all the weights are
exhausted. There are (λ1 + 1)(λ2 + 1) weights in the product representation,
of which λ1 + λ2 − 2k + 1 are removed in the k th step. The process terminates
P
for k = µ, where µk=0 (λ1 + λ2 − 2k + 1) = (λ1 + 1)(λ2 + 1). The solutions of
this equation are µ = λ1 , λ2 and it is evident that termination occurs when
k reaches the lower of λ1 and λ2 . So (λ1 ) ⊗ (λ2 ) = (λ1 + λ2 ) ⊕ (λ1 + λ2 − 2) ⊕
(λ1 + λ2 − 4) ⊕ · · · ⊕ (|λ1 − λ2 |). Recalling the standardization λ = 2j, this is
recognized as the familiar (j1 )⊗(j2 ) = (j1 +j2 )⊕(j1 +j2 −1)⊕· · ·⊕(|j1 −j2 |).
[Note the trivial application of Dynkin’s second highest weight
theorem. A1 has a single simple root with Dynkin index 2. Any
two irreps (λ1 ) and (λ2 ) with λi 6= 0 are connected to the simple
root and there is a chain λ1 → (2) → λ2 . So (λ1 ) ⊗ (λ2 ) contains
the irrep (λ1 + λ2 − 2), as found above.]


2 −1
0

2 −2 
2. The algebra so(7) (or B3 ), has Cartan matrix A =  −1
 and
0 −1
2
Dynkin diagram h h x. This is a rank-3 algebra with irreps (λ, µ, ν) of
dimension d(λ, µ, ν) = (λ + 1)(µ + 1)(ν + 1)(λ + µ + 2)(µ +ν + 2)(λ +µ+ ν + 3)
(2µ + ν + 3)(λ + 2µ + ν + 4)(2λ + 2µ + ν + 5)/720. The building-up process
identifies the adjoint representation as (0, 1, 0), of dimension 21. Consider
the product of irreps (1, 0, 0) ⊗ (0, 1, 0), which has highest weight (1, 0, 0) +
(0, 1, 0) = (1, 1, 0). Note d(1, 0, 0) = 7; d(0, 1, 0) = 21; d(1, 1, 0) = 105, so
the product irrep has d = 147 and there are 42 weights not accounted for
by the highest weight irrep it contains. In the extended Dynkin diagram
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Michael W. Kirson
14. Products of representations
e
5
e
h x there
is a chain (1, 0, 0) → α(1) → α(2) → (0, 1, 0) and α(1) +
α(2) = (1, 1, −2) (the sum of the first two rows of the Cartan matrix.). By
the second highest weight theorem, the product contains the irrep (0, 0, 2),
of dimension 35. There remain 7 weights unaccounted for in the product.
But (0, 1, 0) is the adjoint representation, so the product must contain the
second factor, (1, 0, 0), of dimension 7. All weights are now accounted for —
(1, 0, 0) ⊗ (0, 1, 0) = (1, 1, 0) ⊕ (0, 0, 2) ⊕ (1, 0, 0).
h
Introductory Algebra for Physicists
Michael W. Kirson