Tensor product of representations
In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few.
for all and . Although not every element of is expressible in the form , the universal property of the tensor product operation guarantees that this action is well defined.
In the language of homomorphisms, if the actions of on and are given by homomorphisms and , then the tensor product representation is given by the homomorphism given by
where is the tensor product of linear maps.
One can extend the notion of tensor products to any finite number of representations. If V is a linear representation of a group G, then with the above linear action, the tensor algebra is an algebraic representation of G; i.e., each element of G acts as an algebra automorphism.
Lie algebra representations
The motivation for this definition comes from the case in which and come from representations and of a Lie group . In that case, a simple computation shows that the Lie algebra representation associated to is given by the preceding formula.
Action on linear maps
If and are representations of a group , let denote the space of all linear maps from to . Then can be given the structure of a representation by defining
for all . Now, there is a natural isomorphism
Let denote the endomorphism algebra of V and let A denote the subalgebra of consisting of symmetric tensors. The main theorem of invariant theory states that A is semisimple when the characteristic of the base field is zero.
The general problem
The tensor product of two irreducible representations of a group or Lie algebra is usually not irreducible. It is therefore of interest to attempt to decompose into irreducible pieces. This decomposition problem is known as the Clebsch–Gordan problem.
The SU(2) case
The prototype example of this problem is the case of the rotation group SO(3)—or its double cover, the special unitary group SU(2). The irreducible representations of SU(2) are described by a parameter , whose possible values are
Consider, as an example, the tensor product of the four-dimensional representation and the three-dimensional representation . The tensor product representation has dimension 12 and decomposes as
where the representations on the right-hand side have dimension 6, 4, and 2, respectively. We may summarize this result arithmetically as .
The SU(3) case
In the case of the group SU(3), all the irreducible representations can be generated from the standard 3-dimensional representation and its dual, as follows. To generate the representation with label , one takes the tensor product of copies of the standard representation and copies of the dual of the standard representation, and then takes the invariant subspace generated by the tensor product of the highest weight vectors.
In contrast to the situation for SU(2), in the Clebsch–Gordan decomposition for SU(3), a given irreducible representation may occur more than once in the decomposition of .
As with vector spaces, one can define the kth tensor power of a representation V to be the vector space with the action given above.
The symmetric and alternating square
Over a field of characteristic zero, the symmetric and alternating squares are subrepresentations of the second tensor power. They can be used to define the Frobenius–Schur indicator, which indicates whether a given irreducible character is real, complex, or quaternionic. They are examples of Schur functors. They are defined as follows.
One can define a linear transformation on any vector space with a basis by defining its action on basis elements and extending linearly. Let V be a vector space with basis . The tensor product has a basis . Define an endomorphism (self-map) T of as follows:
Define two subsets of the second tensor power of V:
These are the symmetric square of V and the alternating square of V, respectively. The symmetric and alternating squares are also known as the symmetric part and antisymmetric part of the tensor product.
The second tensor power of a linear representation V of a group G decomposes as the direct sum of the symmetric and alternating squares:
If V has a basis , then the symmetric square has a basis and the alternating square has a basis . Accordingly,
Let be the character of . Then we can calculate the characters of the symmetric and alternating squares as follows: for all g in G,
The symmetric and exterior powers
As in multilinear algebra, over a field of characteristic zero, one can more generally define the kth symmetric power and kth exterior power , which are subspaces of the kth tensor power (see those pages for more detail on this construction). They are also subrepresentations, but higher tensor powers no longer decompose as their direct sum.
- is an irreducible representation of the symmetric group corresponding to a partition of n (in decreasing order),
- is the image of the Young symmetrizer .
The mapping is a functor called the Schur functor. It generalizes the constructions of symmetric and exterior powers:
In particular, as an G-module, the above simplifies to
Tensor products involving Schur functors
Let denote the Schur functor defined according to a partition . Then there is the following decomposition:
where the multiplicities are given by the Littlewood–Richardson rule.
Given finite-dimensional vector spaces V, W, the Schur functors Sλ give the decomposition
The left-hand side can be identified with the ring k[Hom(V, W)] = k[V * ⊗ W] of polynomial functions on Hom(V, W) and so the above also gives the decomposition of k[Hom(V, W)].
Tensor products representations as representations of product groups
Let G, H be two groups and let and be representations of G and H, respectively. Then we can let the direct product group act on the tensor product space by the formula
Even if , we can still perform this construction, so that the tensor product of two representations of could, alternatively, be viewed as a representation of rather than a representation of . It is therefore important to clarify whether the tensor product of two representations of is being viewed as a representation of or as a representation of .
In contrast to the Clebsch–Gordan problem discussed above, the tensor product of two irreducible representations of is irreducible when viewed as a representation of the product group .
- Serre 1977, p. 8.
- Fulton & Harris 1991, p. 4.
- Hall 2015 Section 4.3.2
- Hall 2015 Definition 4.19
- Hall 2015 Proposition 4.18
- Hall 2015 pp. 433–434
- Hall 2015 Theorem C.1
- Hall 2015 Proof of Proposition 6.17
- Serre 1977, p. 9.
- James 2001, p. 196.
- James 2001, Proposition 19.12.
- James 2001, Proposition 19.13.
- James 2001, Proposition 19.14.
- Fulton–Harris, § 6.1. just after Corollay 6.6.
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