This post is going to be an amalgamation of some of the various ideas I’ve had in the last year, I guess. A year ago I thought about writing a post going over some of the representation theory of algebraic groups. This was a fools errand as the year in between taught me.
Instead, I’ve decided to write some posts that give details on the algebraic groups of type A — whatever that means at the moment if you haven’t heard it before. There are a couple of reasons why I want to do this: it’s good to work out some examples, I’ll have computations available for later, I eventually would like to have a pretty comprehensive list for algebraic groups of type ABCDEFG so why not start with A.
Since I’m repurposing an old blog, there might be a bit of heterogeneity in the content below. But I now realize that’s totally fine because this is a blog; it doesn’t have to be a treatise, or a textbook account, or have all the details.
Let’s start out with some vocabulary. The objects I want to talk about are certain group schemes. These are objects which you should think of as being both a scheme, and a group. This is the analog in scheme theory of the concept of a Lie group in, say, complex manifold theory. There are problems with this interpretation from the get-go however. For example: how do you multiply a nonclosed generic point by a closed point, what does it mean to multiply two points of a scheme — given that the underlying set of a product is not the product of the underlying sets, and how does this form a group? This is, in my opinion, one of the biggest differences (or problems) one will first encounter when they try to generalize the idea of a Lie group to a scheme.
So how do we get around this problem? Well, we can generalize our definitions. What are the basic properties of a group? They are the existence of an identity, inverses, and the ability to multiply. Since we can’t use points directly, we’ll have to try to capture these properties with morphisms instead.
(1.1) Definition: A group scheme over a scheme is a group object in the category of schemes over . This means a group scheme over is a scheme with morphisms , , and such that the following equalities hold:
c) (identity); here is the canonical isomorphism provided by the fibered product.
The examples of group schemes this post will focus on are called split semisimple linear algebraic groups of type A. The type A part may be familiar from other areas of Lie theory — it comes from the root data we associate to our algebraic group. When I say algebraic group, I really just mean a group scheme over , where is some field, and being over is shorthand for being over . But, to handle the simplest case first, we only want to deal with finite dimensional objects. This means we also impose a finite type (read as: finite dimensional) assumption when we speak about algebraic groups. Formally, we have just said:
(1.2) Definition: an algebraic group is a group scheme of finite type over a field .
Before I talk about why these definitions give us the ability to think of a scheme as a group, let me talk a little about its implications in the case is an affine algebraic group (when is a group scheme over an arbitrary scheme , the same conclusions could be made locally when is affine over . I won’t work in this generality. The most general situation I’ll consider is when is affine, and is affine over which, in particular, implies is affine).
Because of the equivalence between commutative rings and affine schemes, we get maps going in the other direction, satisfying the reverse conditions of definition 1.1, whenever we have an affine . The ring associated to an affine group scheme is then called a Hopf algebra.
(1.3) Definition: let be a commutative ring. An Hopf algebra is an -algebra together with comultiplication, coinversion, and counit maps: , , which satisfy the relations
c) (identity); here is the canonical isomorphism provided by the tensor product.
We’ll really only need to use Hopf algebras to show some properties of the underlying scheme of the algebraic group we are talking about. Possibly in another post (to be added), I’ll work out an example of how one can work with an algebraic group completely ring/scheme theoretically (but really only a simple one, probably or because I’ve done it before).
Now let’s return to why these properties in particular should capture the group theoretic properties an algebraic group is supposed to have. Often times it’s easier to work directly with the functor defined by the “points” of an algebraic group instead of the algebraic group itself. More generally, every scheme defines a functor , and the group schemes can be identified as those schemes such that this functor factors with the last arrow being the forgetful functor (taking a group to its underlying set).
To see this, look again at the morphisms of definition 1.1. Try to convince yourself that, if these were given on a set instead of a scheme , then they would determine the structure of a group. Now observe, via the functor description of a scheme , the same arrows translate into natural transformations of functors
and so on… If we plug a scheme into these functors, then these really are just maps of sets, and from here you’ve convinced yourself these maps determined a group structure (but now on the set ).
From this we can work more categorically. I’ll summarize the main points of what we will need in the below paragraphs but, if you want to see more detail then take a look at Alex Youcis’ blog here.
Choose a scheme . As I mentioned above, this defines a functor . We can construct a category whose objects are functors from to and whose morphisms are natural transformations of functors. Let’s call such a category and define a morphism of categories as the pair of maps , , and , . With this notation it’s possible to show
(1.4) Proposition: The morphism is a fully faithful embedding.
This is a great result for many reasons. Here are two that we’ll use fairly often. Proposition 1.4 says to give a morhpism of algebraic groups it is equivalent to give a morphism of groups for every scheme . It’s actually incredibly easier to do the latter; it’s easier to check that we have defined a group homomorphism for every scheme than it is to check we have defined a morphism of schemes commuting with their group structures. The second use of prop. 1.4 is: say we define a functor and we can find a scheme such that for all schemes . Then it follows is the only scheme with this property. Said differently: schemes are uniquely determined by the functors they define. We’ll use both of these properties quite frequently.
But, before we continue, we can simplify this even further. One might wonder if we really need to check all schemes for these results to be true (maybe because general schemes are just affine schemes glued together). It turns out that we don’t. It’s sufficient to check these conditions on the affine schemes only. Formally,
(1.5) Lemma: Any functor which factors through is uniquely determined by its restriction to .
Now, with a given scheme , I want to emphasize the scheme and not the functor , so I’ll use (as is common practice) the notation . Other notational conventions I’ve chosen in this blog are as follows. An arbitrary field is denoted , and a finite type -algebra is denoted . I use for base change of along a morphism of affine schemes . I’ve already used this notation but, to avoid writing more than I want to, I’ll often identify and unless I use both simultaneously.
To conclude this section, we’ll end with some examples (some of which we’ll end up considering in the detail of the structure theory of algebraic groups in the following sections).
(1.6) Example: Define the functor by . We say it is represented by the scheme since, on affine schemes , we have
The only nontrivial equality in the above is a result of the universal mapping property of . This allows us to identify any morphism with the image of . We then give the set the group structure so that the functor factors through . This is called the additive group.
I used some formal terminology in the above, so I might as well include it as a definition.
(1.7) Definition: A functor is represented by an object if .
(1.8) Example: Define the functor by . A similar calculation to the above shows is represented by . Note for an affine scheme we find . This functor is called the vector group of dimension n. Most of the time I probably won’t write the subscript . Instead I’ll say of dimension , sticking with the convention from vector spaces.
(1.9) Example: For a vector space , the functor is defined on affines by . When is finite dimensional then the choice of a basis for shows it is represented by the affine scheme where is the determinant of the matrix in the variables (e.g. for the determinant is ). If we’ve chosen a basis we’ll write instead of . To see this functor is represented by , we can proceed as in (Ex. 1.6.):
By the universal property of localization, is the subset of such that is invertible. We give this set the group structure of matrix multiplication. This is the general linear group. The special case is written and called the multiplicative group.
Remark: In the above example, is a group valued functor for any vector space . We can always describe a morphism to, or from, this functor. That is, we can always give morphisms between functors in the target category of (Prop 1.4.) but, we have only shown this will be a morphism of algebraic groups if is finite dimensional and the other functor in the morphism is represented by a scheme.
(1.10) Example: The special linear group is the quotient of at the ideal . The group scheme structure is the same as in example (1.9). In particular, it is a sub-group scheme of .
Algebraic groups – type A
From here on out including any background is pretty much out of the question. I’m going to assume everything I know, and a couple of things I don’t know, about the structure theory of algebraic groups. All of our schemes are going to be over a fixed but arbitrary field and all of our group schemes are going to be affine (these are also called linear; I usually refer to them just as algebraic groups). The starting point for this series of blogs is going to be the following type of theorem.
(2.1) Theorem: split semisimple linear algebraic groups are classified, up to isomorphism, by the pair made of: the root system of and the fundamental group of .
The secondary starting point, and only for this post, is going to be the following lemma.
(2.2) Lemma: is semisimple of type .
This will proved later in this post (above Theorem 2.8). But for now let’s take it on faith and study some of the geometric properties of . There are multiple proofs for the following, and I’m not sure which proof is the cleanest. But here are some proofs which work, at least in the case of .
(2.3) Lemma: is smooth and connected.
Proof. Since is a variety (finite type over a field), it’s flat over its base. This means we only have to prove is geometrically regular to show that its smooth. Since we’ve already seen is an algebraic group (example 1.10), it’s sufficient to show is geometrically reduced. To do this it’s sufficient to show, after base changing to an algebraic closure of that the determinant minus 1 is irreducible. Hopefully you’re a sane person who picked the variables to be outside any power of a transcendence base of the field over its prime field (which was really implicit in my definition of in the first place). The determinant has degree 1 in each of the variables so, if factors as for two polynomials then only one can have a term. Say does and write , . Similarly has degree 1 in , so one of has degree 1 in . It can’t be however, because then there would be a term in the determinant, which is visibly seen to be false by checking the determinant via cofactor expansion. A similar argument works with all the , so has all the terms. But again, this is all of them, as can be seen from expanding the determinant by the top row. We’re done then since implies is constant (as it can contain no variables else the left side has degree 2), and this in turn implies so that and is irreducible.
This also determines the dimension of as the dimension of minus 1. Since is an open subset of an irreducible variety, it has the same dimension as which is . We get, as a corollary,
(2.4) Corollary: has dimension .
Proof. Given above.
Unfortunately that’s about as far as I can go in terms of the geometry of this algebraic group. It would be interesting if one could say more (computing the cohomology ring over has been done, as an example, so we should be able to read off geometric information from there. For arbitrary fields I don’t know what one can say).
In a different direction, we could examine the algebraic structure of .
(2.5) Lemma: the diagonal subgroup variety is a split maximal torus. This can be realized as either or as the subfunctor consisting of diagonal matrices.
Proof. We want to show that is isomorphic with some power of . Using the functor approach this is pretty straightforward, just define an isomorphism by
Hence this torus is split. To see that is maximal, we will show the centralizer of in is itself. To show this, we’ll use the fact that the Lie algebra of the centralizer is equal to the fixed points of the action of on the Lie algebra of . It will turn out that the Lie algebra of the centralizer has dimension , which will complete the proof since the centralizer of a torus is smooth and connected so if we have this implies . The rest is proved in the following lemma.
(2.6) Lemma: the Lie algebra of is isomorphic with the -span of and where and run over the numbers from . The fixed points of the action of on are exactly the vectors .
Proof. The vector represents the matrix with 0’s every except the th row and th column, where there is a 1 instead. Recall the Lie algebra of an algebraic group is defined to be the group-kernel of the map which is given by composing with the natural projection . This is the tangent space at the identity of the group. To compute the Lie algebra of we first compute the Lie algebra for . These are all matrices with values in with invertible determinant and satisfying the condition if then . All such matrices have to be equal to a sum where is an arbitrary matrix of and all matrices of this form appear in the Lie algebra. Multiplying any two matrices of , say and gives
which allows us to define an isomorphism with the vector space of matrices with values in .
We identify with the sub-Lie algebra of consisting of those matrices with determinant 1. If , it’s easy to see the determinant of a matrix is equal . Computing the determinant of such a matrix via cofactor expansion (of the top row) we find with the appropriate minors. All of summands vanish except the first, since can be computed by cofactor expansion along a column containing all terms degree 1 in . By induction this shows is an element of only when which happens if and only if . Under the isomorphism we’ve identified with those matrices having trace 0.
The claim about being spanned by the and where can then be seen as: this is a vector space of dimension , and is the kernel of the map which is surjective. So they must be equal since they have the same dimension.
Finally, we come to the torus action. acts on via conjugation, . We have inclusions for every finite type -algebra and every algebraic group . We also have inclusions . Evaluating the conjugation map on -points gives us then an action . The inclusion and defines the action of on . In short, it’s just conjugation. If is a diagonal matrix, multiplying out shows the claim.
Our next goal is to use this torus to study the root system of (or ). We’ll start by determining the Weyl group for .
(2.7) Lemma: The Weyl group of is isomorphic with .
Proof. We can do this in two ways. I want to do it in both ways because they give different information and both are useful. The first way is directly from the following definition: the Weyl group of an algebraic group with respect to a split maximal torus is defined as the group . This is a constant group scheme so we can identify it with its rational points, or with its points over an algebraic closure so that we could work directly with a quotient of groups. I’ll just give explicit representatives for this group, and defer that this is enough data until after we look at the second method of proof. We work with the maximal torus described in lemma 2.5.
Let be the permutation matrix which transposes the rows and . This isn’t an element in but, it is up to a sign change. Multiply one of the rows of by (the choice doesn’t matter; they are all equivalent modulo ). Let be the group of elements generated by these matrices as runs over the numbers . No two are equivalent matrices mod , by comparing the position of 0’s in the matrices. To see that these are all the necessary representatives of , we will compute this group another way.
Note that the character group of the split maximal torus of is a free abelian group of rank . Under the isomorphism described above, this group is spanned by the elements where goes through under the relation . A base for this lattice is given by where for instance. On the other hand, the cocharacter group is spanned by the elements where is the map with in the th spot and 1’s elsewhere. This can be considered in the lattice spanned by subject to the relation the sum of the coefficients must equal 0. If we call the lattice spanned by modulo and the sublattice of the lattice spanned by subject to the relation the sum of the coefficients is 0, then we have a natural duality between and . This duality is realized by the pairing . Eventually I’ll write that paragraph less miserably.
Anyways, now we want to look at the root and the weight lattice. The root lattice is the sublattice of spanned by the roots. Going back to our calculation of the action of on , we find . Hence the root lattice is spanned by those characters . This has index in but I won’t check this at moment. Via the above pairing, the coroots are exactly the elements , which span . This also shows will be simply connected (once we have shown it is semisimple).
The final piece of this proof is to show the reflections generate . Then, as this group is isomorphic with we will have shown the Weyl group is . We will have also found representatives of the group explicitly since we originally found distinct elements and this shows there can be at most .
Now observe acts as the transposition swapping on the characters . These reflections generate a subgroup of , and it is known the transpositions generate the entire group. So we are done.
That was a lot of work for seemingly not very much reward. But we’re now ready to prove Theorem 2.2. Actually, we could have done the first part of 2.2 at any time but, the second part had to wait until we had the computation of the Weyl group completed. I decided to do them together for whatever arbitrary reason.
Proof of 2.2. That is semisimple follows from the Lie-Kolchin theorem. We change base to an algebraic closure, and then we can assume a solvable subgroup is given as a subgroup of upper-triangular matrices. Since the radical of is normal, it is closed under conjugation so it must also be a subgroup of the lower-triangular matrices. Hence it is diagonal. Let be a matrix contained in the radical of . Then a computation with shows which is diagonal only when . Hence all the entries of are the same. Therefore the radical is contained in the scalar matrices, which for is the group . The only connected subgroup variety of this is the trivial group, which shows is semisimple. To see has root system of type , we should start by finding a base for the given root system. Relative to the Borel subgroup of upper-triangular matrices, the simple roots are the . Taking the inner product gives 0 if , and 2 when . This is known as the root system of type .
I’m going to end this section with some facts about the family of type that I don’t want to prove at the moment (or don’t know how to prove at the moment).
(2.8) Theorem: The algebraic group is the simply connected semisimple algebraic group of type . It is almost simple with center . The adjoint algebraic group of type is . The adjoint algebraic group is simple (I think; at least if it is over a field with greater or equal four elements). Thus, the almost simple algebraic groups of type are the quotients of by subgroups for varying and divisors of .
Homogeneous varieties – type A
This section is, truth be told, the ulterior motive of this whole post. Unfortunately, it’s a broad enough topic I don’t want to include all of the details. Instead, this section will focus on examples. Moreover, all of the examples I’ll work out will be split (as opposed to twisted — what this means is I will consider Grassmannians and Flag varieties instead of their twisted forms, e.g. Severi Brauer varieties will not be discussed).
(3.1) Definition: Let be a vector space over . Define the functor on points as
This is called the Grassmannian of -planes in .
To be more explicit about the defining conditions of this functor, we note that is a -module by its right action. Then is an element of this set if there is a projective -module so that . The restriction maps on the functor are those coming from changing the base.
Note that there is symmetry in the definition. In other words the natural transformation given by sending a summand to its complement defines a bijection on points, hence is naturally an isomorphism. There is also the obligatory mention, .
(3.2) Theorem: The Grassmannian is represented by a scheme, which we also call the Grassmannian, and we also denote by .
Reference. Here’s a reference.
The proofs in the Stacks project are more general than are being considered here. I’ve translated some of the conditions to the case where the base is a field.
(3.3) Definition: Let be a vector space over . Given a strictly increasing sequence of positive integers we define the functor on points by
It’s called the –split partial flag variety of type A.
(3.4) Theorem: For any sequence as above, the functor is represented by a scheme. We give this scheme the same name and notation as the functor it defines.
Proof Sketch. Choose a basis for . Then two flags in are given with respect to this basis, without loss of generality, by and where the are sums in forming a sequence of linearly independent vectors. acts on the first flag, taking . By scaling we can assume the matrix this defines has determinant 1, and is hence in . Picking a point of for this action realizes the flag variety as a quotient of by the isotropy group of this point.
This is a sketch — not a proof. Mainly because one would need to check that the flag variety functor is the quotient of . This might mean adjusting the definition to include extensions by faithfully flat algebras… It’s actually a lot of work, and I don’t really know how to formalize it properly at the moment. That’s why I insist the above is only a sketch. The change won’t really affect us, whichever one it is.
Remark: Grassmannians are special cases of partial flag varieties.
(3.5) Example: Let’s compute the variety for as the quotient by some parabolic subgroup (I know it’s parabolic because Grassmannians are projective — this follows from the Plücker embedding which I won’t go into detail about). Let be the standard basis vector for with 1 in the th position and 0’s elsewhere. A plane in is given by the (span of the) pair and we know the action of on the Grassmannian is transitive. We’ll find then to be the isotropy group . A computation shows
The subgroup of generated by these matrices will then do the trick. To be formal we should actually say the algebraic group defined by the vanishing of the equations .
I want to continue this example by computing the Bruhat decomposition of (for purposes of a later post). The Plücker embedding actually shows embeds in with one defining equation. In particular it has dimension 4. This is helpful as a guide but not strictly necessary. Let’s find a Levi decomposition for . That is, we’ll write where is a reductive group and is a unipotent group.
I claim . In fact, I claim we have a decomposition, for every and every ,
where is the identity, are elements of , , and . To see this, write
, , .
Observe, since we can compute this using block matrices. Setting , , and gives the desired decomposition.
From here it’s easy to see the Weyl group , where is the same split torus we described in section 2. It’s corresponding to the subgroup of generated by the permutations and . The Bruhat decomposition is given then by the recipe
where the union takes place over one representative from each coset of , and is the standard Borel of upper triangular matrices. Since the numerator has 24 elements, and the denominator has 4, we are consequently looking for 6 specific representatives. Here are some choices, one from each coset, in a list:
, , , , , .
Counting inversions of the above permutations we can compute the Bruhat order for these elements. This means we find the length of each permutation to be . From general theory, this calculates the dimensions of the given closures in as . To compute equations defining these invariants seems like too large a task at the moment, so I’ll stop here. This is enough for some purposes.
(3.6) Example: As a second example, we repeat the above computations for the flag variety corresponding to the parabolic subgroup such that . Let represent the standard basis vectors, and consider the flag . Since acts transitively on , computing the isotropy group at will provide us with such a quotient.
But we can skip the calculation of the matrix group , if we want, because is the variety of complete flags. The stabilizer for the standard coordinate flag is just the Borel subgroup of upper triangular matrices, , which has been referred to throughout this whole blog. That is, is the subvariety of corresponding to the vanishing of the functions .
To compute the Bruhat decomposition we can use the entire Weyl group . More precisely, the decomposition appears in the form
The six representatives are the six permutation matrices,
, , , , ,
and the lengths of these elements are .