Evaluation maps provide a tool for understanding how one representation (of a group, say, or of a ring) can appear in another. The basic idea is as follows: suppose that and are representations of a group (over a field , say). How could we tell if appears as a subrepresentation of ? Well, if it does, there will certainly be a -linear -equivariant homomorphism (namely, the inclusion of as a subrepresentation of ). So, it makes sense to look at the -vector space of all -linear -equivariant maps from to . Any such homomorphism may be evaluated on elements of to yield elements of , and so we get a map
precisely, this map is defined by This map is evidently -bilinear, and so tautologically gives rise to a -linear map on the tensor product
defined by . We call this map the evaluation map.
Note that the source and target of this map both have -actions: on the source, there is the action induced by the action on , while on the target, there is the action on . Furthermore, one checks that the evaluation map is -equivariant. Simply compute that . It provides a tautological measurement of the relationship between the representations and . For example, it is non-zero if and only if some non-zero quotient of embeds as a subrepresentation of . By imposing various conditions on and , one can draw various conclusions of differing degrees of strength.
One important point is that the formation of is functorial (equivalently, natural) in and , and so if or have any extra structure, this structure is inherited by We will see applications of this in the examples below.
Basic graduate level algebra, including tensor products and representation theory.
Before we study more general examples of evaluation maps, it will be helpful to study the structure of the source of the evaluation map.
Suppose to begin with that is a finite-dimensional irreducible -representation over an algebraically closed field . If is a -vector space, then the tensor product is equipped with a -action (via the -action on the first factor), and so is naturally a -representaiton over .
What does it look like? Well, if we choose a basis of , then we obtain an isomorphism and hence an isomorphism of -representations . So is just a direct sum of copies of . However, the basis of is not canonical. On the other hand, the -vector space is canonically determined by the representation , as the following result shows.
Proof of Theorem 1. Tensoring with induces a canonical map
which we claim is an isomorphism. To check this, we can compute this map after choosing a basis of . It then becomes identified with the natural map
which is immediately seen to be an isomorphism.
Now since is algebraically closed and if finite-dimensional, Schur's lemma shows that . Since , we obtain the required isomorphism After identifying and via this isomorphism, the evaluation map becomes the identity map , and thus the evaluation map is seen to be an isomorphism, as claimed.□
We have seen that unlike the indexing set , the vector space is canonically determined by . Still, it is reasonable to ask what advantage comes from describing our representation in the form , rather than just explicitly as the direct sum ? Well, here is one advantage: suppose that is any subrepresentation of . Then one can show that is also isomorphic to a direct sum of copies of , say . But the subset need not be in any natural way a subset of . (Just as we may not in general be able to choose the basis of a subspace of a given vector space to be a subset of some particular fixed basis of the entire space.) On the other hand, we have the following result.
Proof of Theorem 2. Let be a subrepresentation. Since (as we remarked above) is isomorphic to a direct sum of copies of , we may write for some set . Thus Theorem 1 shows that the evaluation map is an isomorphism. On the other hand, since we have a corresponding inclusion Hence, if we write and use Theorem 1 to identify ◊ with , we see that we may regard as a subspace of , and that the evaluation map identifies with , as claimed.□
Suppose that and are representation of the group over an algebraically closed field , with irreducible. Then is called the multiplicity space of in . It measures how many independent copies of appears as subrepresentations of . More precisely, we have the following theorem.
Proof of Theorem 3. If is any irreducible subrepresentation of isomorphic to , then there is an isomorphism Thus the image of under the evaluation map is equal to , and thus the image of the evaluation map contains the sum of all subobjects isomorphic to a direct sum of copies of . Thus it contains their sum. Denote this sum by ; it is a subrepresentation of contained in the image of the evaluation map.
On the other hand, is isomorphic to a direct sum of copies of , and so certainly its image under the evaluation map is a sum of copies of subrepresentations isomorphic to . Hence its image is contained in . Consequently, we see that the image of the evaluation map is precisely equal to . We also see that the inclusion is in fact an equality. Thus to complete the proof of the theorem, it suffices to show that is isomorphic to a direct sum of copies of , since Theorem 1 will then imply that the evaluation map
is an isomorphism.
If we let denote the collection of all subrepresentations of that are isomorphic to (thus is an index set labelling all such ), then there is a canonical surjection
Now and thus Theorem 2 shows that the kernel of the above surjection has the form for some subspace . Thus is isomorphic to
and so is indeed isomorphic to a direct sum of copies of . As we already remarked, this completes the proof.□
Suppose that and are two groups.
Proof of Theorem 4. Suppose that is a -subrepresentation of . Thinking of simply as a -subrepresentation for the moment, Theorem 2 shows that has the form for some . Moreover, since is in fact a -subrepresentation of , we see that must be an -invariant subspace of , i.e. an -subrepresentation of . Since is irreducible by assumption, we see that either or . Thus either or , and so is an irreducible -representation, as claimed.□
We now prove the converse of Theorem 4.
Proof of Theorem 5. To begin with, think of just as a -representation. It may no longer be irreducible, but we may find an irreducible -subrepresentation, say . (This is where we use finite-dimensionality.) As above, we construct the evaluation map
Now since is a -representation, it has an action of that commutes with the action of . (To say that is endowed with a -action is just to say that it is endowed with a -action and an -action that commute with one another.) Thus this -action on induces an -action on , i.e. is naturally an -representation. The source of the evaluation map is then a -representation (via the -action on the first factor and the -action on the second factor).
Since is a subspace of the space of all (not necessarily -equivariant) -linear maps from to , it is finite dimensional, and so contains an irreducible -representation, say . The tensor product is a -subrepresentation of and so the evaluation map restricts to give a -equivariant map
We claim that this map is non-zero: if is non-zero, then it is a non-zero map from to (by definition), and thus we may find an element such that . The evaluation map then sends to and so we see that indeed we have a non-zero map.
The preceding example shows that is irreducible subrepresentation of . Thus we have a non-zero map between irreducible subrepresentations of , which must thus be an isomorphism. Since our map is a non-zero -equivariant map, its image is a non-zero -equivariant subrepresentation of . Since was assumed to be irreducible over , it has no non-zero -subrepresentations other than itself, and so our map must in fact be surjective. Since it is non-zero, its kernel is a proper -submodule of . As this is again an irreducible -representation, the only proper subrepresentation that it contains is the zero representation. Thus our map has vanishing kernel, and so is also injective. This completes the proof.□
The proof of Theorem 5 brings out an important point, namely that if is a representation of , and is a -representation, then the space is naturally an -representation. This construction is frequently used (by making a careful choice of ) to create interesting correspondences between representations of a group and of a group .
We illustrate this remark further with some more examples.
Let , the matrix group of invertible -matrices over , and to be the symmetric group for some . The group has a natural representation on , namely the usual action of -matrices on length column vectors. If we take , then permuting the factors in the tensor product gives a representation of on , commuting with the -action. Thus is a -representation, and the map gives a map from (isomorphism classes of) irreducible -representations to (isomorphism classes of) -representations. It turns out that the multiplicity space is irreducible as an -representation (if it is non-zero), and this construction gives a bijection bewteen those irreducible representations of which embed into , and certain irreducible representations of . (If then in fact every irreducible representation of arises in this way.) This is known as Schur-Weyl duality.
Here are two more very briefly sketched examples, both at a much more advanced level than the rest of this article:
In the context of the Langlands program, one can take to be the finite adélic points of some reductive linear algebraic group over , to be an appropriately chosen Galois group, and to be a representation of constructed out of the cohomology of a Shimura variety attached to . One will take to be (the finite part of) an irreducible automorphic representation of the group . The passage from to will then give a construction of Galois representations associated to automorphic representations.
In the general theory of theta functions, one takes and to be the members of a dual reductive pair of groups, and is a certain Weil representation. The resulting passage from -representations to -representations is then referred to as the theta correspondence.