Parameterizing orbits in flag varieties


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1 Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits. We use the parameters to describe the dimension of these orbits, and the natural partial order on them. Many crucial results in algebraic geometry and group theory depend upon knowledge about a certain family of objects: projective spaces, their generalization to Grassmannian varieties, and the further generalization to flag varieties. One way to understand these objects is to study various groups which act upon them and to analyze their orbits. This is the purpose of this article. Now we give a technical statement of the context of this problem. Theorem 1 ([3, 5]). Let G = GL(V ), let X be a maximal rank reductive subgroup of G and P be a proper parabolic subgroup of G. Then X\G/P is finite if and only if one of the following holds. 1. X = GL(V 1 ) GL(V 2 ), 2. X = GL(V 1 ) GL(V 2 ) GL(V 3 ) and P is the stabilizer of a single subspace, 3. P is the stabilizer of subspace of dimension 1, or codimension 1. In this article we parameterize the Xorbits in each of the cases listed in this theorem, and give dimension formulas and closure relations using these parameters. Thus, the present paper gives more precise information about some of the double cosets studied in [3] and [5]. This paper can also be viewed as an extension of the work by Brion and Helminck [2], who parameterized the Xorbits and orbit closures in varieties G/P, where X was assumed to be the fixed points of an involutive automorphism of G. In Theorems 6, 8, 12 and 15 we give parameterizations for the orbits arising in Theorem 1. In Corollaries 9, 13, and 16 we give formulas for the dimension of each orbit, and describe the natural partial order in the set of orbits. Throughout the paper we translate statements about double cosets X\G/P into statements about the collection of Xorbits in G/P, i.e. the Xorbits in a flag variety. We start by giving a little background about these topics. 1
2 1 Background and preliminaries We record some information here that can be found in standard texts, such as [1] and [4]. Let V be a vector space of dimension n over an algebraically closed field of arbitrary characteristic. (Those results that make sense over an arbitrary field are usually true without assuming algebraic closure.) Let GL(V ) be the group of invertible linear transformations on V. Let X be a maximal rank reductive subgroup of GL(V ). Equivalently, there exists a decomposition V = V 1 V s such that X equals a direct product of groups X = GL(V 1 ) GL(V s ). Let P(V ) be the projective space formed on V, i.e. the collection of all 1 spaces in V. For each r {0,..., dim V } let Gr r (V ) be the collection of all rspaces in V. In particular, Gr 0 (V ) is a single point and Gr 1 (V ) = P(V ). A flag F in V is a finite sequence W 1,..., W k of subspaces of V with W i < W i+1 for each i (whenever convenient we will allow W 0 = 0, but we assume that W i 0 for i > 0). We say the flag is full if dim W i = i for each i, and partial if it is not full. Each flag has a corresponding dimension sequence, (dim W i ) i. For each sequence (r i ) i with values in {0,..., n} and such that r i < r i+1 for each i, we let Fl((r i ) i, V ) be the collection of all flags whose dimension sequence equals (r i ) i. In particular Fl(r, V ) = Gr r (V ). Each collection of flags Fl((r i ) i, V ) forms a complete, projective variety. Let P be a parabolic subgroup of GL(V ). By definition, this means that the quotient GL(V )/P is a complete variety. Equivalently, this means that P is the stabilizer in GL(V ) of a flag. The parabolic subgroup uniquely determines the flag, and thus the dimension sequence. Conversely, two parabolic subgroups have the same dimension sequence if and only if they are conjugate. A parabolic subgroup is a Borel subgroup if it equals the stabilizer of a full flag. Equivalently, it is minimal among the parabolic subgroups. Equivalently, it is maximal among the solvable subgroups. Each parabolic subgroup contains a Levi factor L, which is a maximal among the maximal rank reductive subgroups of P. The subgroup L is not uniquely determined by P, but is unique up to P conjugacy. We can choose L as the direct product of groups L = GL( W 1 ) GL( W s ) where W i+1 is chosen as a subspace of W i+1 such that W i+1 = W i W i+1. Note that dim GL(V ) = n 2. Lemma 2. Let G = GL(V ) act on Fl((r i ) i, V ) in the natural manner and let P be the stabilizer of an element of Fl((r i ) i, V ), and let L be a Levi factor of P. 1. The action is transitive. 2. We have an isomorphism of varieties G/P = Fl((r i ) i, V ). 3. dim Fl((r i ) i, V ) = 1 2 (dim G dim L). Part gives a simple way to calculate the dimension of a flag. In particular, one can use this to show that Gr r (V ) = r(n r). Lemma 3 ([3, 3.4]). Let V = Ṽ V s, let π : V Ṽ be the natural projection, let X = X GL(V s ) with X a subgroup of GL(Ṽ ), let W and W be two rspaces 2
3 in V. If (W Ṽ, π(w )) and (W Ṽ, π(w )) are conjugate under X, then W and W are conjugate under X. Corollary 4. If X 1 = GL(V 1 ) then W and W are conjugate under X if and only if dim W V 1 = dim W V 1 and dim π 1 (W ) = dim π 1 (W ). Proof. For x X we have that dim W V 1 = dim xw W 1 and dim π 1 (W ) = dim π 1 (xw ) have the same dimension, which proves one direction. Now we suppose that the equality holds for the indicated dimensions. Note that (W V 1, π 1 (W )) and (W V 1, π 1 (W )) are partial flags. By Lemma 2 we have that these flags are conjugate under X 1, whence the result follows by Lemma 3. 2 Two factor Levi subgroup Let V be ndimensional, fix a decomposition V = V 1 V 2 with X = GL(V 1 ) GL(V 2 ), and let π i : V V i be the natural projection with respect to this decomposition. Let P be the stabilizer of an rspace. For any index i (0 i n) we let Gr i (V ) be the collection of all ispaces in V. Recall that Gr i (V ) is an irreducible, complete variety of dimension i(n i). In what follows we often identify G/P with Gr r (V ) and view X\G/P as the collection of Xorbits in Gr r (V ). Lemma 5. Two rspaces, W and W, are in the same Xorbit if and only if dim W V i = dim W V i for i = 1, 2. Proof. is clear. For we write W i = W V i for i = 1, 2 and similarly for W. We decompose W and W as shown W = W 1 W 2 W W = W 1 W 2 W where W and W are any direct complements. We pick ordered bases respecting these direct sums B = B 1 B 2 B B = B 1 B 2 B These definitions imply that π i is injective on W i W and on W i W. Therefore π i (B i B) and π i (B i B ) are linearly independent sets of the same cardinality. For i = 1, 2 let g i GL(V i ) be an order preserving map taking π i (B i B) to π i (B i B ). Then we have that g i is an order preserving map taking π i B to π i B. Finally, g = (g 1, g 2 ) X takes B to B whence W to W. Theorem 6. The Xorbits on G/P are parameterized by a collection of three part compositions (a 1, a 2, a 3 ) of r where the correspondence is given by Xorbit of W (a 2, a 2, a 3 ) 3
4 with a i = dim W V i (i = 1, 2) and a 3 = r a 1 a 2. A composition (a 1, a 2, a 3 ) of r corresponds to an Xorbit if and only if a i + a 3 n i for i = 1, 2. Note that the component a 3 is somewhat redundant in this parameterization. However, we include a 3 because using it makes the final statement in the previous theorem convenient. Proof. The first statement is an immediate consequence of the previous lemma. Now we prove the last statement. Let W be given, and define a i, π i, W i, W, etc. as above. Since π i is injective when applied to W W i, we have that dim π i ( W W i ) = a 3 + a i and this must be n i. Conversely, let a 1, a 2, a 3 be given. Pick W i V i with dim W i = a i, (i = 1, 2). We now construct W. Let W 3 V 1 be a subspace of dimension a 3, disjoint from W 1 (which is possible since a 3 n 1 a 1 ). Then π1 1 (W 3) is a subspace of V of dimension a 1 + a 3 that contains V 2. Choose a direct complement W of V 2 in π1 1 (W 3). Then W := W 1 W 2 W is the rspace we desire. Corollary 7. Let (W i ) i and (U i ) i be two flags in V with dim W i = dim U i for each i. These flags are in the same Xorbit if and only if for each i 1 and each j = 1, 2 we have dim W i V j = dim U i V j. Proof. The proof is an inductive argument that uses X to zip the two flags together subspace by subspace. Note that U 0 = W 0 = 0. Let j 0 such that there exists x j X with x j U i = W i for all i j. We will show that there exists x X such that x acts as the identity on W i and xx j U j+1 = W j+1, whence the result follows by induction. To simplify notation we replace (U i ) i with (x j U i ) i, and thus assume that U i = W i for all i j. To simplify further, we write U = U i = W i. Thus, we have U < U i+1 and U < W i+1 and wish to construct an element of X than stabilizes U and takes U i+1 to W i+1. For any subspace Z of V, let Z = (Z + U)/U be the corresponding subspace in V/U. Then V = V 1 V 2 and the stabilizer in X of U induces the group X = GL(V 1 ) GL(V 2 ) in GL(V ). Applying Proposition 5 to X, it suffices to show that dim V k U i+1 = dim V k W i+1, for k = 1, 2. (1) Equation (1) holds if and only if dim(v k + U) U i+1 = dim(v k + U) W i+1. By the modular property of the lattice of subspaces we have dim(v k + U) U i+1 = dim U i+1 V K + U, which in turn equals dim U i+1 V k + dim U dim V k U. Since dim U i+1 V k = dim W i+1 V k we see that Equation (1) holds, and we are done. Let P be the stabilizer of a flag in GL(V ) containing k nontrivial subspaces, and let r i equal the dimension of the ith subspace of this flag. 4
5 Theorem 8. The Xorbits in G/P are parameterized by a collection of k 3 arrays (a i,j ) where the ith row, (a i,1, a i,2, a i,3 ) is a composition of r i. The correspondence is given by X.(W i ) i (a i,j ) with a i,j = dim W i V j for j = 1, 2, and a i,3 = r i a i,1 a i,2. Let (a i,j ) be a k 3 array, such that for all i, the ith row is a composition of r i. Then (a i,j ) corresponds to an Xorbit in G/P if and only if a i,j + a i,3 n i for all i and for j = 1, 2, and a i,j a i+1,j for all i and for j = 1, 2, 3. Proof. The statement about parameterization follows immediately from Proposition 7. If (a i,j ) is the label of an orbit, then it is clear that it satisfies the conditions. Now, let (a i,j ) be a k 3 array, satisfying the conditions given in the corollary. We construct a flag (W i ) i such that dim W i = r i, and dim W i V j = a i,j for all i and for j = 1, 2. The construction is inductive, with the main step similar to the inductive step in the proof of Proposition 7. By Theorem 6, there exists W 1 V satisfying dim W 1 V i = a 1,i for i = 1, 2 and dim W 1 = r 1. Inductively, we assume that k 1 and W 1 < < W k are subspaces with a i,j + a i,3 n i for all i k and for j = 1, 2, and a i,j a i+1,j for all i and for j = 1, 2, 3. For each subspace Z < V let Z = (Z + W k )/W k be the corresponding subspace of V = V/W k. Working in V, relative to the decomposition V = V 1 V 2, there exists a subspace W with dim W = r k+1 r k, dim W V i = a k+1,i a k,i for i = 1, 2. Let W k+1 be the lift W of to a subspace of V. Corollary 9. Let (W i ) i be a flag with dimension sequence (r i ) i, with Xorbit X.(W i ) i and let (a i,j ) ij be the label of X.(W i ) i. Then X.(W i ) i has dimension dim Fl((a i,1 ) i, V 1 ) + dim Fl((a i,2 ) i, V 2 ) + dim Fl((a i,3 ) i, V ). If (W i ) i is another flag with dimension sequence (r i ) i and label (b i,j ) then X.(W i ) i X.(W i ) i if and only if b i,j a i,j for all i 1 and for j = 1, 2. In the following proof we make use of elementary methods for constructing open and closed subsets of flag varieties. See [6] for a suitable background. Proof. Let Z 1 = {( W i ) i Fl((r i ) i, V ) dim W i V j a i,j, i, j {1, 2}}. Then Z 1 is closed in Fl((r i ) i, V ), contains (W i ) i, and is stable under X. Therefore X.(W i ) i Z 1 and X.(W i ) i Z 1. This shows that if X.(W i ) i X.(W i ) i, then (W i ) i Z 1 and so b i a i for i = 1, 2. Now we show that if b i,j a i,j for all i and for j = 1, 2 then X.(W i ) i X.(W i ) i. We start by defining varieties Y Y 1 Y 2, and a morphism ϕ : Y 1 Z 1. Let Y = i Gr a i,1 (V 1 ) Gr ai,2 (V 2 ) Gr ai,3 (V ), let Y 1 = {(W i,j ) i,j Y dim(w i,1 +W i,2 +W i,3 ) = r i, for all i}. Note that for (W i,j ) i,j Y we have that dim W i,1 + W i,2 + W i,3 = r i if and only if dim W i,3 (W i,1 W i,2 ) = 0. Thus, Y 1 is an open subset of Y. Let Y 2 = {(W i,j ) i,j Y 1 W i,3 V j = 0, for j = 1, 2}. Then Y 2 is an open subset of Y 1. Since Y is irreducible, we have that Y 1 is as well, and thus Y 2 is dense in Y 1. Define the morphism ϕ : Y 1 Fl((r i ) i, V ) via ϕ((w i,j ) i,j ) = (W i,1 + W i,2 + W i,3 ) i. 5
6 Note that X acts on Y 1 and Y 2, that ϕ is Xequivariant, that (W i ) i ϕ(y 2 ), and that X.(W i ) i = ϕ(y 2 ) (this last by our results showing that the Xorbit of (W i ) i equals all the flags with the same 3 k array of dimensions). Next we will justify each containment in the following sequence X.(W i ) i Z 1 ϕ(y 1 ) ϕ(y 2 ) ϕ(y 2 ) X.(W i ) i, where all closures are taken within Y 1 or Z 1 as appropriate. The first inclusion was proven above. To prove the second inclusion, let (U i ) i Z 1, and write U i = U i,1 U i,2 U i,3 with U i,j = U i V i,j for j = 1, 2. Let U i,j = U i,j K i,j with dim U i,j = a i,j. Then (U i,1, U i,2, K i,1 K i,2 U i,3 ) i Y 1 and U i = ϕ(u i,1, U i,2, K i,1 K i,2 U i,3 ) ϕ(y 1 ). The third inclusion follows from the fact that Y 2 = Y 1, as stated above. The fourth inclusion is an elementary statement in point set topology. The fifth inclusion follows from the fact, stated above, that X.(W i ) i = ϕ(y 2 ). Since (W i ) i Z 1, this shows that (W i ) i X.(W i ) i. Finally, we have dim X.(W i ) i = dim X.(W i ) i = dim Y 2 = dim Y, which proves the statement in the Corollary about dim X.(W i ) i. 3 Three factor Levi subgroup Throughout this section we have V = V 1 V 2 V 3 and X = GL(V 1 ) GL(V 2 ) GL(V 3 ). Let π i : V V i and π ij : V V ij be the natural projections. Lemma 10. Let X = X 1 X 2 X 3 with X i = GL(V i ), V = V 1 V 2 V 3 and let W and W be two rspaces. Then W and W are in the same Xorbit if and only if (W (V 1 V 2 ), π 12 (W )) and (W (V 1 V 2 ), π 12 (W )) are in the same X 1 X 2 orbit. Proof. This is an immediate corollary of Lemma 3, where s = 3 and X = X 1 X 2 = GL(V 1 ) GL(V 2 ). Corollary 11. With the above notation, W and W are in the same Xorbit if and only if dim W V i = dim W V i for 1 i 3, and dim W (V i V j ) = dim W (V i V j ) for 1 i < j 3. Proof. We will combine Corollary 7 and Lemma 10. We need two equations to show that the two partial flags in V 1 V 2 given in Lemma 10 have the same dimensions, and then we need four equations to show that the spaces in these flags give the same dimensions when intersected with V 1 and V 2. Thus, we see 6
7 that that W and W are in the same Xorbit if and only if dim W (V 1 V 2 ) = dim W (V 1 V 2 ), dim π 12 (W ) = dim π 12 (W ), dim(w (V 1 V 2 )) V 1 = dim(w (V 1 V 2 )) V 1, (2) dim(w (V 1 V 2 )) V 2 = dim(w (V 1 V 2 )) V 2, dim π 12 (W ) V 1 = dim π 12 (W ) V 1, dim π 12 (W ) V 2 = dim π 12 (W ) V 2. It is easy to show that dim π 12 (W ) = dim W dim W V 3 and (W (V 1 V 2 )) V 1 = W V 1 and π 12 (W ) V 1 = π 1 (W (V 2 V 3 )). Note that π 1 (W (V 1 V 3 )) = dim W (V 1 V 3 ) dim(w (V 1 V 3 ) (V 2 V 3 )) = dim W (V 1 V 3 ) dim W V 3. Thus, Equations (2) are equivalent to dim W (V 1 V 2 ) = dim W (V 1 V 2 ), dim W dim W V 3 = dim W dim W V 3, dim W V 1 = dim W V 1, dim W V 2 = dim W V 2, dim W (V 1 V 3 ) dim W V 3 = dim W (V 1 V 3 ) dim W V 3, dim W (V 2 V 3 ) dim W V 3 = dim W (V 2 V 3 ) dim W V 3. Note that dim W = dim W, so we can eliminate these terms from the second equation. We can also eliminate terms involving W V 3 and W V 3 from the last two equations. Theorem 12. With the above notation, the Xorbits on G/P = Gr r (V ) are parameterized by a collection of 6tuples (a 1, a 2, a 3, a 12, a 13, a 23 ) where the correspondence is given by Xorbit of W (a 1, a 2, a 3, a 12, a 13, a 23 ) with a i = dim W V i, (1 i 3), and a ij = dim W (V i V j ), (1 i < j 3). A 6tuple (a 1, a 2, a 3, a 12, a 13, a 23 ) of nonnegative integers corresponds to an Xorbit if and only if a ij a i n j and r a ij n k for 1 i < j 3 and k i, j. Proof. The statement about parameterizing follows immediately from Corollary 11. Given an rspace W, define a i and a ij, as above. Then Theorem 6, applied three times, to subspaces of V 1 V 2, V 1 V 3 and V 2 V 3, gives a ij a i n j for 1 i < j 3. To see that r a ij n k, note that r a ij = dim π k (W ). Conversely, let a i and a ij be given. We will construct an rspace W with appropriate dimensions. The proof is similar to the analogous part of Theorem 6, but more complicated because the dimensions of the various subspaces are interrelated. 7
8 Choose subspaces W i V i with dim W i = a i. For j = 2, 3, choose W 1j V 1 of dimension a 1j a 1 a j, disjoint from W 1 (which is possible since a 1j a j n 1 ). Then π 1 1 V 1 V j (W 1j ) is a subspace of V 1 V j of dimension a 1j a 1 that contains W j. Choose a direct complement W 1j of W j in π 1 1 V 1 V j (W 1j ). Then W 1, W 2, W 3, W 12, W 13 are independent and dim W 1j = a 1j a 1 a j. Now let K be a subspace of V 1 disjoint from π 1 (W 1 W 12 W 13 ) of dimension r a 23 (which is possible since r a 23 n 1 ). Note that π 1 1 (K) contains V 2 V 3. Let W23 be a direct complement of V 2 V 3 in π 1 1 (K). Then dim W23 = r a 23 and W23 is independent from W 1, W 2, W 3, W 12, W 23. We set W = W 1 W 2 W 3 W 12 W 23 W23. Corollary 13. Let X.W be the Xorbit of W Gr r (V ) with corresponding label (a 1, a 2, a 3, a 12, a 13, a 23 ). Then the dimension of X.W is given by dim X.W = 3 a i (n i a i ) + i=1 1 i<j 3 (a ij a i a j )(n i + n j a ij + a i + a j ). If X.W is the Xorbit of another subspace W Gr r (V ) labelled by the 6tuple (b 1, b 2, b 3, b 12, b 13, b 23 ). Then X.W X.W if and only if b i a i and b ij a ij for all i and j. Proof. We define the subvariety F 3 Gr ai (V i ) i=1 1 i<j 3 Gr aij (V i V j ) consisting of all those 6tuples (W 1, W 2, W 3, W 12, W 13, W 23 ) such that W ij V i = W i and W ij V j for 1 i < j 3. As in the proof of Corollary 9, we can define a map from ϕ : F 1 X.W where F 1 is an open subset of F consisting of those 6tuples with dim(w 12 +W 13 +W 23 ) = dim W. In particular, this leads to dim X.W = dim F, and the statement about containment of orbits follows similarly as in the proof of Corollary 9. Thus, we omit some details concentrate on finding dim F. The condition W ij V i = W i can be split into two conditions: W i W ij and dim W ij V i a i. The second condition is open. Thus, we define F consisting of 6tuples (W 1, W 2, W 3, W 12, W 13, W 23 ) with W i W ij and W j W ij for 1 i < j 3, and we calculate dim F. Let ρ be the projection of F to the coordinates (W 1, W 2, W 12 ). The image of ρ is irreducible (we describe it in a moment), and so we can apply the standard result about dimensions of fibres, domain and image: over an open subset the dimension of the domain equals the dimension of each fibre plus the dimension of the image [6]. The image of ρ is isomorphic to a set of triples (W 1, W 2, W ) 8
9 where W W 12 satisfies dim W = a 12 a 1 a 2 and W V i = 0 for i = 1, 2. This is the set of triples studied in Corollary 9 (in Corollary 9, the notation a 3 equaled dim W a 1 a 2 ; here we would replace a 3 with a 12 a 1 a 2 ). Therefore dim ρ(f ) = a 1 (n 1 a 1 ) + a 2 (n 2 a 2 ) + (a 12 a 1 a 2 )(n 1 + n 2 a 12 + a 1 + a 2 ). The fibres of ρ consist of triples (W 3, W 13, W 23 ) subject to the remaining conditions. We can break these triples down further using projections and fibres. The W 3 component leads to Gr a3 (V 3 ). The W 13 component leads to the set of all subspace W 13 of dimension a 13 a 1 a 3 in V 1 V 3 /W 1 W 3. This has dimension dim Gr a13 a 1 a 3 (V 1 V 3 /W 1 W 3 ) and W 23 leads to dim Gr a23 a 2 a 3 (V 2 V 3 /W 2 W 3 ). Adding the dimensions of all these varieties, we get the desired result. 4 Many factor Levi subgroups We use similar notation as in the previous sections. Note, that the two cases under consideration in this section are essentially equivalent: the action of X on V has a dual action on the dual space V, and 1spaces in V correspond to (n 1)spaces in V. Thus, we restrict our attention to the case where P is the stabilizer of a 1space. Let V = V 1 V m, X = GL(V 1 ) GL(V m ) and let π i : V V i be the natural projection. Lemma 14. Two 1spaces W and W are in the same Xorbit if and only if for all i we have dim W j i V j = dim W j i V j. This condition is also equivalent to dim π i W = dim π i W for all i. Proof. This follows from Lemma 3 and induction on m. Theorem 15. The Xorbits on G/P are parameterized by a collection of m tuples (a 1,..., a m ) where a i = dim W ( j i V j). An mtuple (a 1,..., a m ) corresponds to an Xorbit if and only if a i {0, 1} for all i, with at least one a i not equal to 1. Proof. The first statement follows from Lemma 14. For the second statement, if W is a 1space, then it is clear that a i equals 0 or 1. Conversely, suppose we are given an mtuple of (a 1,..., a m ) with each a i equal to 0 or 1. For each i, we pick v i V i with the condition v i = 0 a i 0. Then let W be the 1space spanned by v v m. Corollary 16. Let W G/P have label (a 1,..., a m ) and orbit X.W. i 1,..., i s be those indices such that a i = 0. Then Let dim X.W = dim P(V i1 V is ). If W G/P has label (b 1,..., b m ) and orbit X.W then X.W X.W if and only if b i a i 9
10 Proof. This result follows in the same manner as Corollaries 9 and 13, but it is perhaps easier to understand if we translate the statements to projections instead of intersections. The X.W orbit is dense in V i1 V is since the projection of W to each V i is nontrivial. This implies that the closure of X.W will contain X.W if W has nontrivial projections to a subset of V i1,..., V is. References [1] Armand Borel, Linear Algebraic Groups, 2nd ed, GTM, vol. 126, SpringerVerlag, [2] Michel Brion and Aloysius G. Helminck, On orbit closures of symmetric subgroups in flag varieties, Canad. J. Math. 52 (2000), no. 2, [3] W. Ethan Duckworth, A classification of certain finite double coset collections in the classical groups, Bull. London Math. Soc. 36 (2004), [4] Larry C. Grove, Classical groups and geometric algebra, GSM, vol. 39, American Mathematical Society, [5] Robert M. Guralnick, Martin W. Liebeck, Dugald Macpherson, and Gary M. Seitz, Modules for algebraic groups with finitely many orbits on subspaces, J. Algebra 196 (1997), no. 1, [6] Joe Harris, Algebraic Geometry: a first course, GTM, vol. 133, Springer Verlag,
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