This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few.
, the universal property of the tensor product guarantees that this action is well-defined.
is the tensor product of linear maps.
[3] One can extend the notion of tensor products to any finite number of representations.
The motivation for the use of the Kronecker sum in this definition comes from the case in which
In that case, a simple computation shows that the Lie algebra representation associated to
[5] For quantum groups, the coproduct is no longer co-commutative.
However, the permutation map remains an isomorphism of vector spaces.
denote the space of all linear maps from
The main theorem of invariant theory states that A is semisimple when the characteristic of the base field is zero.
The tensor product of two irreducible representations
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).
then decomposes as follows:[7] Consider, as an example, the tensor product of the four-dimensional representation
copies of the dual of the standard representation, and then takes the invariant subspace generated by the tensor product of the highest weight vectors.
Over a field of characteristic zero, the symmetric and alternating squares are subrepresentations of the second tensor power.
Define two subsets of the second tensor power of V, These are the symmetric square of V,
[11] The second tensor power of a linear representation V of a group G decomposes as the direct sum of the symmetric and alternating squares: as representations.
In the language of modules over the group ring, the symmetric and alternating squares are
Then we can calculate the characters of the symmetric and alternating squares as follows: for all g in G, As in multilinear algebra, over a field of characteristic zero, one can more generally define the kth symmetric 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.
The Schur–Weyl duality computes the irreducible representations occurring in tensor powers of representations of the general linear group
It generalizes the constructions of symmetric and exterior powers: In particular, as a G-module, the above simplifies to where
denote the Schur functor defined according to a partition
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 on Hom(V, W ), k[Hom(V, W )] = k[V * ⊗ W ], and so the above also gives the decomposition of k[Hom(V, W )].
act on the tensor product space
, we can still perform this construction, so that the tensor product of two representations of
It is therefore important to clarify whether the tensor product of two representations 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