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.

Definition

Group representations

If are linear representations of a group, then their tensor product is the tensor product of vector spaces with the linear action of uniquely determined by the condition that
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

If and are representations of a Lie algebra, then the tensor product of these representations is given by the map given by
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
as vector spaces; this vector space isomorphism is in fact an isomorphism of representations.
The trivial subrepresentation consists of G-linear maps; i.e.,
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.

Clebsch–Gordan theory

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—or its double cover, the special unitary group SU. The irreducible representations of SU are described by a parameter, whose possible values are
Let us take two parameters and with. Then the tensor product representation then decomposes as follows:
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, 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, in the Clebsch–Gordan decomposition for SU, a given irreducible representation may occur more than once in the decomposition of.

Tensor power

As with vector spaces, one can define the th tensor power of a representation 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.
Let be a vector space. Define an endomorphism of as follows:
It is an involution, and so is an automorphism of.
Define two subsets of the second tensor power of :
These are the symmetric square of and the alternating square of , respectively. The symmetric and alternating squares are also known as the symmetric part and antisymmetric part of the tensor product.

Properties

The second tensor power of a linear representation of a group decomposes as the direct sum of the symmetric and alternating squares:
as representations. In particular, both are subrepresenations of the second tensor power. In the language of modules over the group ring, the symmetric and alternating squares are -submodules of.
If 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 in,

The symmetric and exterior powers

As in multilinear algebra, over a field of characteristic zero, one can more generally define the th symmetric power and th exterior power, which are subspaces of the th tensor power. They are also subrepresentations, but higher tensor powers no longer decompose as their direct sum.
The Schur–Weyl duality computes the irreducible representations occurring in tensor powers of representations of the general linear group. Precisely, as an -module
where
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
where. Moreover, the multiplicity may be computed by the Frobenius formula. For example, take. Then there are exactly three partitions: and, as it turns out,. Hence,

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 of polynomial functions|ring k = k of polynomial functions on Hom and so the above also gives the decomposition of k.

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.