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 the map given by , where is the identity endomorphism. This is called the Kronecker sum, defined in Matrix addition#Kronecker_sum and Kronecker product#Properties. The motivation for the use of the Kronecker sum in 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 prototypical 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
(The dimension of the representation is then .) 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(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 .
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.