Tensor product of representations

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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[edit]

Group representations[edit]

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

[1][2]

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.[3]

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[edit]

If and are representations of a Lie algebra , then the tensor product of these representations is given by the map given by[4]

.

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.[5]

Action on linear maps[edit]

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;[2] this vector space isomorphism is in fact an isomorphism of representations.[6]

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[edit]

The general problem[edit]

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[edit]

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

(The dimension of the representation is then .) Let us take two parameters and with . Then the tensor product representation then decomposes as follows:[7]

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[edit]

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.[8]

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[edit]

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[edit]

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:

This transformation is independent of the chosen basis. It is an involution (it is its own inverse), and so is an automorphism (self-isomorphism) of .

Define two subsets of the second tensor power of V:

These are the symmetric square of V and the alternating square of V, respectively.[9] The symmetric and alternating squares are also known as the symmetric part and antisymmetric part of the tensor product.[10]

Properties[edit]

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 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 .[11]

If V has a basis , then the symmetric square has a basis and the alternating square has a basis . Accordingly,

[12][9]

Let be the character of . Then we can calculate the characters of the symmetric and alternating squares as follows: for all g in G,

[13]

The symmetric and exterior powers[edit]

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.

The Schur–Weyl duality computes the irreducible representations occurring in tensor powers of representations of the general linear group . Precisely, as an -module

where

  • 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

where . Moreover, the multiplicity may be computed by the Frobenius formula (or the hook length formula). For example, take . Then there are exactly three partitions: and, as it turns out, . Hence,

Tensor products involving Schur functors[edit]

Let denote the Schur functor defined according to a partition . Then there is the following decomposition:[14]

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[edit]

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 .

See also[edit]

Notes[edit]

  1. ^ Serre 1977, p. 8.
  2. ^ a b Fulton & Harris 1991, p. 4.
  3. ^ Hall 2015 Section 4.3.2
  4. ^ Hall 2015 Definition 4.19
  5. ^ Hall 2015 Proposition 4.18
  6. ^ Hall 2015 pp. 433–434
  7. ^ Hall 2015 Theorem C.1
  8. ^ Hall 2015 Proof of Proposition 6.17
  9. ^ a b Serre 1977, p. 9.
  10. ^ James 2001, p. 196.
  11. ^ James 2001, Proposition 19.12.
  12. ^ James 2001, Proposition 19.13.
  13. ^ James 2001, Proposition 19.14.
  14. ^ Fulton–Harris, § 6.1. just after Corollay 6.6.

References[edit]

  • Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
  • Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666.
  • James, Gordon Douglas (2001). Representations and characters of groups. Liebeck, Martin W. (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 978-0521003926. OCLC 52220683.
  • Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402 .
  • Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. Springer-Verlag. ISBN 978-0-387-90190-9. OCLC 2202385.