A few years ago, I gave a talk at NCSU on some work I had done on Littlewood-Richardson numbers, cluster algebras and such things. For the first half hour or so, I outlined the basic results I would be using about the representation theory of the group . Afterwards, I had a number of grad students thank me for this. So I’m going to try to turn that into a blog post (and enlarge it a little). The goal here is not to give you any proofs; rather, I want to get to the main results, show you how they connect and, above all, how to actually write down the representations of .
is, of course, the group of complex matrices with invertible determinant. We want to classify the linear representations of , meaning we want to find group homomorphisms from . Some examples: We have the trivial representation, where and every matrix in is mapped to the identity. We have the determinant representation, where again, and is mapped to . We have the standard representation, where and is mapped to itself. We have the dual of the standard representation, which is given in coordinates by . We have the symmetric representations, where acts on the symmetric powers of the standard representation, and the anti-symmetric or exterior representations, where acts on the anti-symmetric or wedge powers of the standard representation. If you have studied almost any field of math, I think it is safe to say that you have frequently encountered these examples; hopefully, that will suggest to you that classifying all representations of is a worthwhile problem.
Now, there is a technical point we have to get out of the way. If all we ask for is a group homomorphism, there are far too many. For example, we can use complex conjugation to get maps like . More confusingly, we can use other field automorphisms of , to get highly discontinuous maps. Another way things can get odd is that, as a group, has a lot of automorphisms (at least, if you believe in the axiom of choice), we could compose with any of these. Moreover, we could take any of these weird examples and tensor them with a normal example to get more weird ones. So we will want to come up with a rule that excludes these examples and limits us to the more algebraic examples of the preceding paragraph.
I’m an algebraic geometer, so my preferred fix is to require that the map is an algebraic map. This means that every entry of be given by a polynomial in the entries of and . (In most of the examples I gave of representations, the entries of are polynomials in the entries of , but in the dual representation we need to have as well.) One of the ways that I would defend my preferred choice is to point out that many seemingly different choices give the same result. You could also require that be holomorphic, and you would get the exact same set of maps. You could (this is the physicists’ choice) study the unitary group, , and require your maps to be continuous (or, alternately, smooth); then every representation you found would extend uniquely to an algebraic representation of . You could take the definition that I originally gave, using algebraic maps, and run it over any field of characteristic zero, and the description I will give in this post will still be correct. For that reason, I am trying to choose my notation to avoid mentioning the complex numbers whenever possible, although being perfectly consistent about this would be more of a pain than I think it is worth.
The first thing you need to know about is that it is what is called a reductive group. That means that any finite dimensional representation of splits as a direct sum of irreducible representations. (An irreducible representation is a representation which contains no subrepresentations other than and itself.) This splitting is unique in the appropriate sense, which takes a little effort to state correctly. For an example of a group that is not reductive, consider the group of complex numbers under addition; the representation can not be split into irreducible representations. So, we will be done if we can understand the irreducible representations of .
It isn’t just finite dimensional representations that are tamed by reductiveness. If is any algebraic variety (of finite type over ) and is an algebraic action of on , then it is easy to show that the coordinate ring, , of has an ascending filtration by finite dimensional representations. Reductiveness lets us split this filtration, so we get that is an infinite direct sum of finite dimensional irreducible representations. In particular, we can consider the action of on itself. Better, we can consider the action of on itself, with one copy of acting from the left and the other on the right. Explicitly, the action is .
(Those inverses are not where you expect them because the correspondence between and is contravariant. I’d advise you not to worry too hard about this.)
The Peter-Weyl theorem: the coordinate ring of , as a representation, is .
The sum runs over the isomorphism classes of irreducible representations of . I prefer to rewrite the summand as . (Note that the Wikipedia link, at least today, states this result in the analytic setting rather than the algebraic one. This is just another example of how which category you work in doesn’t matter very much for reductive groups.) If you have seen some representation theory, then you should be familiar with Peter-Weyl theorem in the setting of finite groups, where it states that the regular representation decomposes in this manner.
Let’s take an easy example. If , then is . The irreducible representations of are indexed by the integer ; the representation is .
There are two good ways to use the Peter-Weyl theorem to describe the representation of .
The first, analogous to the use of characters in the representation theory of finite groups, is to look at those functions on which are invariant under action of the diagonal. (I just realized that the word “diagonal” is ambiguous. I am talking about functions which are invariant under the subgroup of .)
On the one hand, these are the functions such that . In other words, functions which depend only on the conjugacy class of their input. Now, the diagonalizable matrices are dense in , so any function on is determined by its values on the diagonalizable matrices. Furthermore, if is to be a conjugacy invariant function, then the value of on diagonalizable matrices is determined by its value on diagonal matrices. So we can describe such an by giving its value on . Finally, note that is conjugate to ; more generally, conjugation can reorder the entries of a diagonal matrix arbitrarily. So must be a symmetric function of the . Moreover, since we are working with algebraic maps and algebraic functions throughout, must be a symmetric Laurent polynomial in the . Conversely, any symmetric Laurent polynomial gives a conjugacy invariant polynomial function on .
On the other hand, from the presentation , we see that each irreducible representation contributes a single basis element to the diagonal invariants in . (Namely, the identity map from to itself.) Explicitly, is the trace of acting on . For example, the standard representation gives us and the dual of the standard representation gives us . The symmetric functions are called Schur functions; by the discussion of the previous paragraph, the Schur functions are a -basis for the symmetric Laurent polynomials.
Now, there is an obvious basis for the symmetric Laurent polynomials, namely, the monomial symmetric functions. There is one of these for each decreasing -tuple of integers, . We just take the sum where ranges over the permutations of .
So, on some intuitive level, we expect that there is one irreducible representation for each such -tuple . The idea which makes this precise is the idea of high weight vectors. If is a representation of then is called a high weight vector if is fixed by every upper triangular matrix with ‘s on the diagonal.
Theorem 1: In every irreducible representation, up to scaling, there is a unique high weight vector.
For example, in the standard representation, is the high weight vector. It is also easy to check that the diagonal matrices send high weight vectors to themselves. So, if is an irreducible representation and its high weight vector, then for some .
Theorem 2: In the above setting, we always have . Conversely, for decreasing -tuple of integers, there is a unique irreducible representation such that the high weight vector transforms in this manner.
Now you know a set which is in bijection with the irreducible representations. But I promised I’d tell you how to write them down. The way to do this is the second good way to use the Peter-Weyl Theorem. We introduce the notation for the group of upper triangular matrices with ‘s on the diagonal. So Theorem 1 tells us that, if we take invariants in , we get .
Now invariant elements of wind up corresponding to functions such that whenever . Some obvious functions with this property are the coordinate functions on the bottom row of our matrix: namely, , …, on . More subtly, any bottom justified minor is invariant under left multiplication by . For example, if and we write coordinates on as
then the determinant is left -invariant.
In fact, these determinants, along with , generate the ring of left -invariants in . There are many proofs of this, my favorite is Theorem 14.11 of Miller-Sturmfels.
So, the ring generated by these determinants is . How do you write down an individual ? You can extract this from everything I’ve said, but I’ll just tell you the answer. is the vector space spanned by products which use determinants of size , determinants of size and so forth, up to determinants of size . (Note that may be negative.)
(The statement that this recipe works is essentially the Borel-Weil Theorem in the algebraic category. More specifically, let be the group of upper triangular matrices. We showed that the space of functions on which transform in a certain way under the diagonal matrices is the vector space . Borel-Weil says that the sections of a certain line bundle on is the same vector space. The equivalence between these two viewpoints, compared to what came before, is not bad.)
Let’s wrap up with an example: we’ll take and . We must look at products which use one determinant of size one and one determinant of size two. Using the coordinates on above, we need to look at the vector space spanned by
, , , , , , , and .
These are 9 products here, but they span a vector space of dimension because
Choose 8 of these products to get a basis, and it is no trouble at all to write down . In particular, it is easy to compute the Schur function ; it is
As a final remark, there is no completely natural way to choose 8 of the 9 products above, and this problem only becomes worse as grows. There are some nice ways though, and one of the simplest is described in Corollary 14.9 of Miller-Sturmfels. An alternate way is the subject of my recent note with Kyle Petersen and Pavlo Pylyavskyy.