Configurations of Linear Subspaces and Rational ... - Project Euclid

Configurations of Linear Subspaces and Rational Invariants Dm i tr i Z ai tsev

1. Introduction Let Gr n,d (C) denote the Grassmannian of all d-dimensional linear subspaces in C n , and let GL n (C) × (Gr n,d (C)) s → (Gr n,d (C)) s be the canonical diagonal action. Dolgachev [DB] posed the following question: Is the quotient Gr n,2 (C) s/GL n (C) (e.g., in the sense of Rosenlicht) always rational? Recall that a Rosenlicht quotient of an algebraic variety X acted on by an algebraic group G is an algebraic variety V together with a rational map X → V whose generic fibers coincide with the G-orbits. Such quotients always exist and are unique up to birational isomorphisms [R]. In the sequel all quotients will be assumed of this type. An algebraic variety Q is rational if it is birationally equivalent to P m with m = dim Q. We answer the above question in the affirmative by applying the rationality of the quotient (GL 2 (C))2/GL 2 (C), where GL 2 (C) acts diagonally by conjugations (see [P1] and surveys [B; D]). Theorem 1.1. For all positive integers n and s, the quotient (Gr n,2 (C)) s/GL n (C) is rational. Equivalently, the field of rational GL n (C)-invariants on (Gr n,2 (C)) s is pure transcendental. The statement of Theorem 1.1 has been recently proved by Megyesi [M] in the case n = 4 and by Dolgachev and Boden [DB] in the case of odd n. Their proofs are independent of the present one. More generally, we show the birational equivalence between (Gr n,d (C)) s/ GL n (C) and certain quotients of matrix spaces. Let GL n (C) × (GL n (C)) s → (GL n (C)) s be the action defined by (g, M1, . . . , Ms ) 7→ (gM1 g −1, . . . , gMs g −1). The first main result of the present paper consists of the following two statements. Theorem 1.2. (1) Let s and d be arbitrary positive integers, and let n = rd for some integer r > 1. Then (Gr n,d (C)) s/GL n (C) is birationally equivalent to (GL d (C)) k/ GL d (C), where Received August 3, 1998. Revision received December 14, 1998. Michigan Math. J. 46 (1999).

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k=

(r − 1)(s − r − 1) if s > r + 1, 1

else.

(2) Let s be an arbitrary positive integer. Let n = (2r + 1)e and d = 2e for some integers r and e. Then (Gr n,d (C)) s/GL n (C) is birationally equivalent to (GL e (C)) k/GL e (C), where   2r − 4 + (4r − 2)(s − r − 2) if r > 1, s > r + 1, k = 2(s − 4) if r = 1, s > 4,  1 else. If n is even then Theorem 1.1 follows from the first part of Theorem 1.2 and from the rationality of the quotient of (GL 2 (C)) s by GL 2 (C) (see Procesi [P1]); if n is odd then it follows from the second part of Theorem 1.2. Formanek [F1; F2] proved the rationality of (GL n (C)) s/GL n (C) for n = 3, 4. Using his result together with Theorem 1.2, we obtain the following. Theorem 1.3. For every positive integer s, we have: (1) (Gr n,3 (C)) s/GL n (C) is rational for n = 0 (mod 3); (2) (Gr n,4 (C)) s/GL n (C) is rational for n = 0 (mod 2); (3) (Gr n,5 (C)) s/GL n (C) is rational for n = 3 (mod 6); (4) (Gr n,8 (C)) s/GL n (C) is rational for n = 4 (mod 8); We refer the reader to [BS] for similar equivalences of stable rationalities (an algebraic variety V is stable rational if V × P m is rational for some m). We also refer to [GP1; GP2] for the classification of quadruples of linear subspaces of arbitrary dimensions and their invariants. In our situation, however, all rational invariants of the quadruples (i.e., the case s = 4) are constant unless r = 2 in the first part of Theorem 1.2. Our method is based on constructing certain normal forms for our algebraic group actions. We call an algebraic variety acted upon algebraically by an algebraic group G a G-space. A G-subspace is a G-invariant locally closed algebraic subvariety of X. We use the standard notation Gs := { gs : g ∈ G } and GS := { gs : g ∈ G, s ∈ S }, where S ⊂ X is an arbitrary subset. Definition 1.1. We say that (S, H ) is a normal form for (X, G) if H ⊂ G is a subgroup and S ⊂ X is a H -subspace such that the following hold: (1) GS is Zariski dense in X; (2) Gs ∩ S = Hs for all s ∈ S. Clearly these conditions guarantee the birational equivalence of the quotients X/G and S/H. In Sections 3 and 4 we construct certain normal forms that are isomorphic to the spaces of matrices as in Theorem 1.2. Then, in Section 5, we use these normal forms for explicit computations of generators of the fields of rational invariants in each case of Theorem 1.2. This method also has applications to biholomorphic automorphisms of nonsmooth bounded domains, where the configurations of linear subspaces appear

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naturally as collections of tangent subspaces to the so-called characteristic webs. We refer the reader to [Z] for further details.

2. Notation s For brevity we write GL n and Grn,d for GL n (C) and (Gr n,d (C)) s , respectively. The actions on the products will be assumed diagonal unless otherwise specified. For E and E 0 vector spaces with dim E ≤ dim E 0 , denote by Grd (E ) the Grassmannian of all d-dimensional subspaces of E, by GL(E ) the group of linear automorphisms of E, and by GL(E1, E2 ) the space of all linear embeddings of E1 into E2 . A G-space X is homogeneous (resp. almost homogeneous) if G acts transitively on X (on a Zariski dense subset of X).

3. The Case d Divides n s In this section we study the diagonal GL n -action on Grn,d with n = rd for some r integer r. Clearly, the set of all r-tuples (V1, . . . , Vr ) ∈ Grn,d such that C n = V1 ⊕ · · · ⊕ Vr is Zariski open and GL n -homogeneous. This can be reformulated in terms of normal forms as follows. r is such that Lemma 3.1. Suppose that r ≥ 2, s ≥ r, and (E1, . . . , Er ) ∈ Grn,d n C = E1 ⊕ · · · ⊕ Er . Define s : (V1, . . . , Vr ) = (E1, . . . , Er ) } S1 = S1(s) := { (V1, . . . , Vs ) ∈ Grn,d

and the group H1 := GL(E1 ) × · · · × GL(Er ) ⊂ GL n . Then (S1, H1 ) is a normal s , GL n ). form for (Grn,d Now fix the splitting C n = E1 ⊕ · · · ⊕ Er as before. Let V ∈ Gr n,d be such that its projection on each Ei (i = 1, . . . , r) is bijective. In this case we say that V is in general position with respect to (E1, . . . , Er ). Clearly the subset of all V, which are in general position, is Zariski open in Gr n,d . Thus, for every i = 2, . . . , r, the projection Vi of V on E1 × Ei is a graph of a linear isomorphism ϕ i : E1 → Ei . We claim that this correspondence between elements V ∈ Gr n,d and (r −1)-tuples of linear isomorphisms (ϕ 2 , . . . , ϕ r ) ∈

r Y

GL(E1, Ei )

i=2

is birational. Lemma 3.2.

Under the assumptions of Lemma 3.1, define r Y GL(E1, Ei ) → Gr n,d , 8: i=2

8(ϕ 2 , . . . , ϕ r ) := { (z ⊕ ϕ 2 (z) ⊕ · · · ⊕ ϕ r (z)) : z ∈ E1 }, Q and the H1 -action on ri=2 GL(E1, Ei ) by

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((g1, . . . , gr ), (ϕ 2 , . . . , ϕ r )) 7→ (g 2 B ϕ 2 B g1−1, . . . , gr B ϕ r B g1−1). Then 8 is birational and H1 -equivariant. The image of 8 consists of all subspaces V ∈ Gr n,d that are in general position with respect to (E1, . . . , Er ). Proof. Since 8 defines a system of standard coordinates on Gr n,d , it is birational. The equivariance and the statement on the image are straightforward. Q Since ri=2 GL(E1, Ei ) is H1 -homogeneous, we have the following. Corollary 3.1. Under the assumptions of Lemma 3.1, let S10 (s) ⊂ S1(s) be the Zariski open subset, where Vr+1 is in general position with respect to (E1, . . . , Er ). Then H1 acts transitively on S10 (r + 1). This proves statement 1 of Theorem 1.2 for s ≤ r + 1. Suppose now that s > r + 1 and that Er+1 ∈ Gr n,d is in general position with respect to (E1, . . . , Er ). Lemma 3.3. Define S 2 = S 2 (s) := { (V1, . . . , Vs ) ∈ S1(s) : Vr+1 = Er+1 } and the group H 2 := GL(E1 ) with the action on S 2 defined by the homomorphism H 2 → H1, where

−1 g 7→ (g, ϕ 2 B g B ϕ −1 2 , . . . , ϕ r B g B ϕ r ),

(ϕ 2 , . . . , ϕ r ) := 8−1(Er+1).

s , GL n ). Then (S 2 , H 2 ) is a normal form for (Grn,d

Proof. By Corollary 3.1, H1 S 2 is Zariski open in S1 . By straightforward calculations, (S 2 , H 2 ) is a normal form for (S1, H1 ) and hence, by Lemma 3.1, a normal s , GL n ). form for (Grn,d We now let V ∈ Gr n,d be one more subspace in general position with respect to (E1, . . . , Er ). Define (ψ 2 , . . . , ψr ) := 8−1(V ). Clearly the map r Y

GL(E1, Ei ) → GL(E1 ) r−1,

(1)

i=2

(ψ 2 , . . . , ψr ) 7→

(ϕ −1 2

B ψ2 , . .

. , ϕ r−1

B ψr )

is one-to-one, where (ϕ 2 , . . . , ϕ r ) = 8−1(Er+1) is fixed. Moreover, we obtain the following lemma. Lemma 3.4.

Define K: GL(E1 ) r−1 → Gr n,d by K(χ 2 , . . . , χ r ) := 8(ϕ 2 B χ 2 , . . . , ϕ r B χ r ),

and define the H 2 -action on X2 by (g, (χ 2 , . . . , χ r )) 7→ (g B χ 2 B g −1, . . . , g B χ r B g −1). Then K is birational and H 2 -equivariant. The image of K consists of all subspaces V ∈ Gr n,d that are in general position with respect to (E1, . . . , Er ).


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Proof. The map K is birational as the composition of 8 and the map (1). The other statements are straightforward. Corollary 3.2.

Under the assumptions of Lemma 3.3, define 9 : S 2 (s) → (GL(E1 ) r−1) s−r−1,

9(E1, . . . , Er+1, Vr+2 , . . . , Vs ) := (K −1(Vr+2 ), . . . , K −1(Vs )). Then 9 is birational and H 2 -equivariant. Finally, using Lemma 3.3, we obtain the following. s is almost GL n -homogeneous if Corollary 3.3. For n = rd, the space Grn,d s , GL n ) that and only if s ≤ r + 1. If s > r + 1, there is a normal form for (Grn,d (r−1)(s−r−1) , GL d ). is isomorphic to (GL d

This implies the first part of Theorem 1.2.

4. The Case n = (2 r + 1)e and d = 2 e r We start with an r-tuple (V1, . . . , Vr ) ∈ Grn,d . Clearly the subset of all r-tuples that form a direct sum is both Zariski open and GL n -homogeneous. Fix an r-tuple (E1, . . . , Er ) in this subset and suppose that s > r. A straightforward calculation yields the following.

Lemma 4.1.

Define

s : (V1, . . . , Vr ) = (E1, . . . , Er ) } S1 = S1(s) := { (V1, . . . , Vs ) ∈ Grn,d

and the group H1 := { g ∈ GL n : g(Ei ) = Ei for all i = 1, . . . , r } with the s , GL n ). diagonal action on S1 . Then (S1, H1 ) is a normal form for (Grn,d Now we wish to parameterize the d-dimensional linear subspaces in C n with respect to (E1, . . . , Er ). We say that V is in general position with respect to (E1, . . . , Er ), if dim W = e, where W := V ∩ (E1 ⊕ · · · ⊕ Er ) and the projection of W on each Ei (i = 1, . . . , r) is injective. The subset of all V ∈ Gr n,d that are in general position is clearly Zariski open. We first give another description of the subspaces W ∈ Gr e (E1 ⊕ · · · ⊕ Er ) that are in general position with respect to (E1, . . . , Er ). For this, let A i = A i (V ) ∈ Gr e (Ei ) be the projections of W and let ϕ i = ϕ i (V ) ∈ GL(A1, Ei ) be the linear isomorphisms whose graphs are equal to the projections of W on E1 ⊕ Ei (i = 2, . . . , r). Denote by X = X(E1, . . . , Er ) the space of all tuples (A, ϕ 2 , . . . , ϕ r ), where A ∈ Gr e (E1 ) and ϕ i ∈ GL(A, Ei ), with the standard structure of a quasiprojective variety. Lemma 4.2. Define 8 : X → Gr e (E1 ⊕ · · · ⊕ Er ), 8(A, ϕ 2 , . . . , ϕ r ) := { (z ⊕ ϕ 2 (z) ⊕ · · · ⊕ ϕ r (z)) : z ∈ A },

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and the H1 -action on X by (g, (A, ϕ 2 , . . . , ϕ r )) 7→ (g(A), g B ϕ 2 B g −1, . . . , g B ϕ r B g −1). Then 8 is birational and H1 -equivariant. The image of 8 consists of all subspaces W ∈ X that are in general position with respect to (E1, . . . , Er ). Proof. The proof is straightforward. Corollary 4.1. Let S10 (s) ⊂ S1(s) be the Zariski open subset of all s-tuples V = (V1, . . . , Vs ) such that Vr+1 is in general position with respect to (E1, . . . , Er ). Then H1 acts transitively on S10 (r + 1). Proof. It follows from the general position condition that W(V ) := Vr+1 ∩ (E1 ⊕ · · · ⊕ Er ) ∈ Gr e (E1 ⊕ · · · ⊕ Er ), in the notation of Lemma 4.2. Let V, V 0 ∈ S10 (r + 1) be arbitrary elements. Since X is H1 -homogeneous, there exists g1 ∈ H1 such that g1(W(V )) = W(V 0 ). Then 0 and g 2 |(E1 ⊕ · · · ⊕ Er ) = id. By there exists g 2 ∈ GL n with g 2 (Vr+1) = Vr+1 construction, g 2 ∈ H1 and the proof is finished. Corollary 4.2.

s For s ≤ r + 1, Grn,d is almost GL n -homogeneous.

Let s > r +1 and Er+1 ∈ Gr n,d be in general position with respect to (E1, . . . , Er ). As a direct consequence of Corollary 4.1, we have the following lemma. Lemma 4.3. Define S 2 := { (V1, . . . , Vs ) ∈ S1 : Vr+1 = Er+1 } and H 2 := { g ∈ GL n : g(Ei ) = Ei for all i = 1, . . . , r +1 }, with the diagonal action on S 2 . Then s , GL n ). (S 2 , H 2 ) is a normal form for (Grn,d For the sequel we suppose that the subspaces E1, . . . , Er+1 are fixed. Define (A, ϕ 2 , . . . , ϕ r ) := 8−1(Er+1 ∩ (E1 ⊕ · · · ⊕ Er )). Let V ∈ Gr n,d be one more subspace that is in general position with respect to (E1, . . . , Er ). Clearly, V is not uniquely determined by its e-dimensional intersection Z1(V ) := V ∩ (E1 ⊕ · · · ⊕ Er ). However, using Er+1, we can consider another intersection Z2 (V ) := V ∩ (E1 ⊕ · · · ⊕ Er−1 ⊕ Er+1). Then the map Z := (Z1, Z2 ) : Gr n,d → Gr e (E1 ⊕ · · · ⊕ Er ) × Gr e (E1 ⊕ · · · ⊕ Er−1 ⊕ Er+1) is birational with the inverse Z −1 : (W, W 0 ) 7→ W + W 0 . Using the construction of X and 8 for the tuple (E1, . . . , Er−1, Er+1) instead of (E1, . . . , Er ), we obtain Y := X(E1, . . . , Er−1, Er+1) and the H 2 -equivariant birational map 9 : Y → Gr e (E1 ⊕ · · · ⊕ Er−1 ⊕ Er+1). Combining 8, 9, and Z, we obtain the following.


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Corollary 4.3. The composition := Z −1 B (8, 9) : X × Y → Gr n,d is birational and H 2 -invariant. In the following we treat the cases r > 1 and r = 1 separately. 4.1. The case r > 1. For V ∈ Gr n,d , set (B, ψ 2 , . . . , ψr ) := 8−1(V ∩ (E1 ⊕ · · · ⊕ Er )) ∈ X, −1

(C, χ 2 , . . . , χ r−1, χ r+1) := 9 (V ∩ (E1 ⊕ · · · ⊕ Er−1 ⊕ Er+1)) ∈ Y.

(2) (3)

In order to construct a smaller normal form, fix (A0, ϕ 20 , . . . , ϕ r0 )∈X, A00 ∈Gr e (E1 ), 0 ∈ GL(A00 , Er+1) such that E1 = A ⊕ A0 , A00 ∩ A = A00 ∩ A0 = {0}, Ei = and χ r+1 0 (A00 ). ϕ i (A) ⊕ ϕ i0(A0 ) for all i = 2, . . . , r, and C n = E1 ⊕ · · · ⊕ Er ⊕ χ r+1 Lemma 4.4.

In the notation just described, define

0 }. S := { V ∈ Gr n,d : B = A0 , C = A00 , ψ 2 = ϕ 20 , . . . , ψr = ϕ r0 , χ r+1 = χ r+1

Then H 2 S is Zariski open in Gr n,d . Proof. For V ∈ Gr n,d generic, E1 = A ⊕ B. Hence there exists a g ∈ GL(E1 ) with g|A = id and g(B) = A0 . Clearly g extends to an isomorphism from the action by H 2 . Without loss of generality, B = A0 . Similarly, there exists an isomorphism g ∈ H 2 such that g|ϕ i (A) = id and g B ψi = ϕ i0 on A0 for all i = 2, . . . , s. Thus we may assume that ψi = ϕ i0 . By Lemma 3.1, we may also assume that C = A00 . Again, for V generic, C n = E1 ⊕ · · · ⊕ Er ⊕ χ r+1(A00 ), where χ r+1(A00 ) ⊂ Er+1. Hence there exists an isomorphism g ∈ H 2 such that g|(E1 ⊕ · · · ⊕ Er ) = 0 . This proves the lemma. id and g B χ r+1 = χ r+1 Let α ∈ GL(A, A0 ) be the isomorphism whose graph is A00 with respect to the splitting E1 = A ⊕ A0 . Define β : A → A00 by β(z) := z ⊕ α(z). For every i = 2, . . . , r, consider ξi := (ϕ i × ϕ i0 )−1 B χ i B β ∈ GL(A, A ⊕ A0 ) and its components [ξi ]1 ∈ GL(A) and [ξi ] 2 ∈ GL(A, A0 ). Define the rational map 4 : S → GL(A)2(r−2), 4: V 7→ ([ξ 2 ]1, α −1 B [ξ 2 ] 2 , . . . , [ξr−1]1, α −1 B [ξr−1] 2 ), where S is defined in Lemma 4.4. Let g ∈ GL(A) be arbitrary. Using the isomorphisms α : A → A0 , ϕ i : A → 0 0 B β : A → χ r+1 (A00 ), ϕ i (A) and ϕ i0 : A0 → ϕ i0(A0 ) for i = 2, . . . , r, and χ r+1 we can extend g canonically to an isomorphism from the action by H 2 . This extension defines a canonical GL(A)-action on C n and therefore on S (as defined in Lemma 4.4). We compare this action with the diagonal GL(A)-action on GL(A)2(r−2) by conjugations as follows.

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Lemma 4.5. equivariant.

The map 4 : Gr n,d → GL(A)2(r−2) is birational and GL(A)-

Proof. The inverse of 4 is given by 4−1(λ 2 , . . . , λ r−1, δ 2 , . . . , δr−1) = (x, y), where : X × Y → Gr n,d is birational (by Corollary 4.3) and x := (A0 , ϕ 20 , . . . , ϕ r0 ), y := (A00 , (ϕ 2 × ϕ 20 ) B (λ 2 , α B δ 2 ) B β −1, . . . , 0 0 ) B (λ r−1, α B δr−1) B β −1, χ r+1 ). (ϕ r−1 × ϕ r−1

The equivariance is straightforward. In the following corollary we use the space S as defined in Lemma 4.4. s : Vr+2 ∈ S } and conCorollary 4.4. Define S 3 (s) := { (V1, . . . , Vs ) ∈ Grn,d sider the diagonal GL(A)-action on S 3 . Then (S 3 , GL(A)) is a normal form for s (Grn,d , GL n ). r+2 is almost GL n Corollary 4.5. If d = 2e and n = (2r + 1)e, the space Grn,d homogeneous if and only if r = 2.

Now let V ∈ Gr n,d be one more subspace. Using the previously fixed data, we associate with V a tuple of 4r − 2 linear automorphisms of A as follows. In the notation following (2) and (3), let τ1 ∈ GL(A, A0 ) and τ 2 ∈ GL(A, A0 ) be the linear isomorphisms with the graphs B and C, respectively. As above, let α ∈ GL(A, A0 ) be the isomorphism whose graph is A00 . For j = 1, 2, set σj := α −1 B τj ∈ GL(A) and τj0(z) := z + τj (z) (τ10 : A → B, τ 20 : A → C ). For i = 2, . . . , r, ζi := (ϕ i × ϕ i0 )−1 B ψi B τ10 ∈ GL(A, A ⊕ A0 ); for i = 2, . . . , r − 1, θ i := (ϕ i × ϕ i0 )−1 B χ i B τ 20 ∈ GL(A, A ⊕ A0 ). We write [ζi ]1, [θ i ]1 ∈ GL(A) and [ζi ] 2 , [θ i ] 2 ∈ GL(A, A0 ) for the corresponding components with respect to the splitting E1 = A ⊕ A0 . Then we obtain the linear automorphisms of A as x i := [ζi ]1 and y i := α −1 B [ζi ] 2 for i = 2, . . . , r; z i := [θ i ]1, t i := α −1 B [θ i ] 2 for i = 2, . . . , r − 1. In order to deal with the remainder term χ r+1, we define ϕ r+1 ∈ GL(A, C n ) by ϕ r+1(z) := z ⊕ ϕ 2 (z) ⊕ · · · ⊕ ϕ r (z). It follows from the construction that ϕ r+1(A) ∈ Gr e (Er+1) and Er+1 = ϕ r+1(A) ⊕ 0 (A00 ). Using this splitting we define the remainder isomorphisms χ r+1 0 )−1 B χ r+1 B τ 20 ∈ GL(A, A ⊕ A0 ), θr := (ϕ r+1 × χ r+1

z r := [θr ]1, and t r := α −1 B [θr ] 2 ∈ GL(A).


Lemma 4.6.

195

Let 2 : Gr n,d → GL(A)4r−2 be given by 2: V 7→ (σ1, σ 2 , x 2 , y 2 , z2 , t 2 , . . . , xr , yr , z r , t r ).

Then 2 is birational and GL(A)-equivariant. Proof. In the foregoing notation, the inverse 2−1 can be calculated as follows: τj0 := α B σj + id,

B := τ10 (A),

C := τ 20 (A),

ψi = (ϕ i × ϕ i0 ) B (x i , α B y i ) B (τ10 )−1 for i = 2, . . . , r, χ i = (ϕ i × ϕ i0 ) B (z i , α B t i ) B (τ 20 )−1 for i = 2, . . . , r − 2, 0 ) B (z r , α B t r ) B (τ 20 )−1, χ r+1 = (ϕ r+1 × χ r+1

V = ((B, ψ 2 , . . . , ψr ), (C, χ 2 , . . . , χ r−1, χ r+1)). The equivariance is straightforward. Corollary 4.6.

If s ≥ r + 2 then the space (S 3 , GL(A)) is isomorphic to , GL e ). (GL 2r−4+(s−r−2)(4r−2) e

Proof. The required birational isomorphism is given by (V1, . . . , Vs ) 7→ (4(Vr+2 ), 2(Vr+3 ), . . . , 2(Vs )),

(4)

where 4 is birational by Lemma 4.5 and 2 is birational by Lemma 4.6. This implies the second part of Theorem 1.2 in the case r > 1. 4.2. The Case r = 1 In this case we have n = 3e. Recall that we fixed E1, E2 ∈ Gr n,d in general position such that dim A = e, where A := E1 ∩ E2 . Choose another subspace E3 ∈ Gr n,d such that dim A1 = dim A2 = e, where Aj := Ej ∩ E3 for j = 1, 2. Recall that we defined H 2 ⊂ GL n to be the stabilizer of both E1 and E2 . Denote by H 3 ⊂ H 2 the stabilizer of E3 . Then H 3 = GL(A) × GL(A1 ) × GL(A2 ) with respect to the splitting C n = A ⊕ A1 ⊕ A2 . The following is straightforward. Lemma 4.7. Suppose that s ≥ 3 and define S 3 = S 3 (s) := { (V1, . . . , Vs ) ∈ S 2 : s , GL n ). V 3 = E3 }. Then (S 3 , H 3 ) is a normal form for (Grn,d Now we choose Bj ∈ Gr e (Ej ) such that Bj ∩ A = Bj ∩ Aj = {0} for j = 1, 2. Then each Bj can be seen as the graph of an isomorphism ϕj ∈ GL(A, Aj ). On the other hand, for V ∈ Gr n,d generic, the subspaces Cj := V ∩ Ej are graphs of isomorphisms ψj ∈ GL(A, Aj ). Define g = (id, ψ1 B ϕ 1−1, ψ 2 B ϕ −1 2 ) ∈ GL(A ⊕ A1 ⊕ A2 ). Then g ∈ H 3 and g(Bj ) = Cj for i = 1, 2; in particular, V = B1 + B 2 . Together with Lemma 3.3 this proves the following.

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Lemma 4.8.

Suppose that s ≥ 4. Define

S 4 = S 4 (s) := { (V1, . . . , Vs ) ∈ S 3 : V4 ∩ E1 = B1, V4 ∩ E2 = B 2 } and H 4 := GL(A). Define the H 4 -action on S 4 via the homomorphism H4 → H3,

g 7→ (g, ϕ 1 B g B ϕ 1−1, ϕ 2 B g B ϕ −1 2 ).

s , GL n ). Then (S 4 , H 4 ) is a normal form for (Grn,d

As a special case of Corollary 3.3, we obtain the following. s is almost GL 3e -homogeneous if and only if s ≤ Lemma 4.9. The space Gr3e,2e , GL e ), 4. If s ≥ 5 then there exists a normal form that is isomorphic to (GL 2(s−4) e where GL e acts diagonally by conjugations.

This implies the second part of Theorem 1.2 in the case r = 1.

5. Computation of Rational Invariants 5.1. The Case n = rd s we represent the eleIn order to compute the rational GL rd -invariants of Grrd,d s ments of Grrd,d by the equivalence classes of rd × ds matrices M ∈ C rd×sd , where the equivalence is taken under the right multiplication by GLsd . Then the diagonal s corresponds to the left multiplication on C rd×sd . We start GL rd -action on Grrd,d with 2d × 2d matrices. Define A11 A12 −1 D (5) := A11A−1 D : GL 2d → GL d , 21 A22 A12 . A21 A22

Then D is a rational map that is invariant under the right multiplication by GL d × GL d and the left multiplication by {e} × GL d , where e ∈ GL d is the unit. Moreover, D is equivariant with respect to the left multiplication by GL d × {e}. More generally, let M ∈ C rd×sd be a rectangular matrix, where r and s are arbitrary positive integers. We split M into d × d blocks A ij (i = 1, . . . , r, j = 1, . . . , s). Then, for every i = 2, . . . , r and j = 2, . . . , s, define A11 A1j rd×sd → GL d , D ij (M ) := D . D ij : C A i1 A ij Similarly to D, each D ij is a rational map that is invariant under the right multiplication by GLsd and under the left multiplication by {e} × GLr−1 d . Finally, we construct rational maps that are invariant under the left multiplication by the larger group GL rd . For this, we assume s > r + 1; otherwise, the left GL rd -multiplication is almost homogeneous (see Corollary 3.3) and hence all rational invariants are constant. Then every matrix M ∈ C rd×sd from a Zariski open subset can be split into two blocks: M = (AB),

A ∈ GL rd , B ∈ C rd×(s−r)d .


197

Define ϕ(M ) := A−1B ∈ C rd×(s−r)d and Gij (M ) := D ij (ϕ(M )) for i = 2, . . . , r and j = 2, . . . , s − r. In particular, for d = 1, r = 2 and s = 4, we obtain the classical double ratio: 1 1 1 1 z − z3 z2 − z 4 =D 2 G22 z1 z2 z3 z 4 z1 − z3 z1 − z 4 =

z2 − z3 z2 − z 4 : . z1 − z3 z1 − z 4

By the invariance of D ij , every Gij is invariant under the left multiplication by GL rd and under the right multiplication by {e} × GLs−1 d . Moreover, by construction, Gij is equivariant with respect to the subgroup GL d × {e} ⊂ GL d × GLs−1 d , where we take the right action on C rd×sd and the conjugation on the image space GL d . We therefore obtain the following rational (GL rd × GLsd )-invariants: Iαβ := Tr(Gα1 β1 · · · Gα k β k ) = Tr((Dα1 β1 B ϕ) · · · (Dα k β k B ϕ)),

(6)

where α ∈ {2, . . . , r}k and β ∈ {2, . . . , s − r}k are arbitrary multi-indices. Using a result of Procesi [P2], we show that the invariant field is actually generated by these traces of monomials. Theorem 5.1. Let d, r, s be arbitrary positive integers such that s > r +1. Then the field of rational (GL rd × GLsd )-invariants of rd × sd matrices is generated by the functions Iαβ , where α ∈ {2, . . . , r}k and β ∈ {2, . . . , s − r}k and where k < 2d . Proof. Set n := rd as before. We write the spaces E1, . . . , Er+1, Vr+2 , . . . , Vs ∈ Gr n,d as in Lemma 3.3 in the form of an rd × sd matrix with d × d blocks as   E 0 ··· 0 E E ··· E  0 E · · · 0 E V 2,r+2 · · · V 2,s  (7) M :=  .. ..  .. ,  .. .. . . . .. .. . . . . . . . 0 0 · · · E E Vr,r+2 · · · Vr,s where E denotes the identity matrix of the size d × d. Let S˜ 2 ⊂ C rd×sd be subspace of all matrices of the form (7). Clearly E1, . . . , Er+1 fulfill the assumptions of Lemma 3.3. It then follows from the lemma that (S˜ 2 , GL d ) is a normal form for (C rd×sd , GL rd × GLsd ), where GL d acts on S˜ 2 as the diagonal subgroup of GL rd × GLsd (i.e., each d × d block is conjugated by the same matrix). By the definition of a normal form, it is sufficient to prove that the field of invariants of (S˜ 2 , GL d ) is generated by restrictions of the Iαβ . Let M be given by (7). By the obvious calculations,

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198



E E ϕ(M ) =   .. . E

and hence

E

V 2,r+2 .. . Vr,r+2

··· ··· .. . ···

 E V 2,s  ..   .

(8)

Vr,s

Iαβ (M ) = Tr(Vα1,r+β1 · · · Vα k ,r+β k ).

(9)

By a theorem of Procesi [P2], the polynomial invariants of (r − 1)(s − r − 1)tuples (Vij ) with respect to the diagonal GL d -conjugations are generated by the monomials of the form (9) with k < 2 d . Since all points of (S˜ 2 , GL d ) are semistable (see [MFK]), the categorical quotient S˜ 2 //GL d exists and is given by these monomials. Then the rational invariants on (S˜ 2 , GL d ) are pullbacks of rational functions on S˜ 2 //GL d , and the proof is finished. 5.2. The Case n = 3e and d = 2e s In this case the elements of Grn,d are represented by the equivalence classes of 3e×2se with 3e × 2e blocks M1, . . . , Ms . As before, we are lookmatrices M ∈ C ing for rational invariants with respect to left multiplications by GL 3e and right multiplications by GLsd . We start with the case of 3-blocks. Define the rational map ϕ

A1 B1

A2 B2

ϕ: C 3e×6e → C 3e×6e , A3 E0 E0 E0 := B3 B1 A−1 B 2 A−1 B 3 A−1 1 2 3   E 0 E 0 E 0 =  0 E 0 E 0 E , c1 d1 c 2 d 2 c 3 d 3

(10)

where A1, A2 , A3 ∈ GL d and B1, B 2 , B 3 ∈ C e×2e and where E 0 ∈ GL 2e and E ∈ GL e denote the identity matrices. We see the corresponding subspaces E1, E2 , E3 ∈ Gr n,d as graphs of linear maps given by the matrices (c1 d1 ), (c 2 d 2 ), and (c 3 d 3 ), respectively. Our first goal will be to compute the intersections A := E1 ∩ E2 and Aj := Ej ∩ E3 (j = 1, 2). For this, we set Ã −1 −1 −1 ! c3 d3 c2 d2 E c1 d1 x1 x 2 x 3 , , ∈ C e×3e. := y1 y 2 y 3 c2 d2 c1 d1 c3 d3 E Then x1 c1 + y1 d1 = x1 c 2 + y1 d 2 = E and hence the 3e × e matrix   x1  y1  E represents the intersection E1 ∩ E2 . Similarly the 3e × e matrices     x3 x2  y 2  and  y 3  E E


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represent the intersections E3 ∩ E1 and E2 ∩ E3 , respectively. Equivalently, the 3-tuple (E1, E2 , E3 ) can be represented by the matrix   x1 x 2 x 3 x1 x 2 x 3  y1 y 2 y 3 y1 y 2 y 3 . (11) E E E E E E Now take the general matrix M = (M1, . . . , Ms ) ∈ C 3e×2es . Following the construction of Section 4.2, we bring A, A1, and A2 (i.e., the matrix (11)) to a normal form. For this, consider the square matrix   x1 x 2 x 3 H(M ) :=  y1 y 2 y 3  E E E and multiply M by H(M )−1 : H(M )−1M 

E :=  0 0

0 E 0

0 0 E

E 0 0

0 E 0

0 0 E

C14 C24 C34

D14 D 24 D 34

··· ··· ···

Next we normalize the 3e × 2e blocks:    −1 E C1i D1i  C2i D 2i  C1i D1i = 0 C2i D 2i C3i D 3i α 2i−1

C1s C2s C3s

 D1s D 2s . D 3s

(12)

 0 E  α 2i

and define σ 2i−1 := α 2i−1α 7−1 and

σ 2i := α 2i α8−1 for i = 5, . . . , s.

Comparing this construction with Section 4.2, we conclude that the matrices σj (j = 9, . . . , 2s) represent exactly the 2(s − 4) matrices in the isomorphic normal , GL e ). Using the theorem of Procesi [P2] and arguments similar form (GL 2(s−4) e to those used in Section 5.1, we obtain the following. Theorem 5.2. Let e and s be arbitrary positive integers such that s > 4. Then the field of rational (GL 3e × GLs2e )-invariants of 3e × 2es matrices is generated by the functions Jα := Tr(σα1 · · · σα k ), where α ∈ {9, . . . , 2s}k and k < 2e . 5.3. The Case n = (2r + 1)e and d = 2e s is represented by a matrix M ∈ C (2r+1)e×2se Here an element (V1, . . . , Vs ) ∈ Grn,d with s blocks M1, . . . , Ms of the size (2r + 1)e × 2e. Again, the GL n -invariants s correspond to (GL n × GLs2e )-invariants on C (2r+1)e×2se . Similarly to on Grn,d the previous paragraph, we start with a special case of a (2r + 1)e × (2r + 2)e matrix and compute representatives of the intersections Er+1 ∩ (E1 ⊕ · · · ⊕ Er ),

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200

Er+2 ∩ (E1 ⊕ · · · ⊕ Er ), and Er+2 ∩ (E1 ⊕ · · · ⊕ Er−1 ⊕ Er+1) as in Section 4. We start by normalizing the n × 2e blocks as in (10): 0 ··· E0 E0 E · · · E0 A1 · · · As := = . (13) ϕ B1 · · · Bs B1 A−1 · · · Bs A−1 C1 · · · Cs s 1 In order to compute the first intersection Er+1 ∩ (E1 ⊕ · · · ⊕ Er ), we consider the corresponding system of linear equations:     X1 X1 . 0 0  .  0  (C1 . . . Cr ) ...  = Cr+1Xr+1, = E Xr+1, (E . . . E ) . Xr

Xr

where each Xi is a 2e × e block. Solving from the first equation Xr+1 = X1 + · · · + Xr and substituting this into the second, we obtain   X1 ((C1 − Cr+1) . . . (Cr − Cr+1)) ...  = 0. Xr

This is a system of (2r + 1)e × e equations with 2re × e variables. Hence, for C1, . . . , Cr+1 in general position, this system has a solution that can be represented by rational 2e × e block functions Xi = Xi (M )

(i = 1, . . . , r).

Comparing this with the construction of Section 4, we see that the blocks 0 E (Xi ) (i = 1, . . . , r), Ci represent the subspaces A, ϕ 2 (A), . . . , ϕ r (A). For our normalization ( Lemma 4.4) 0 , which come from the we also need the subspaces A0 , ϕ 20 (A0 ), . . . , ϕ r0 (A0 ) and χ r+1 other intersections. For them we can also solve the corresponding linear systems and obtain rational 2e × e block functions Yi (M ) such that the blocks 0 E (Yi (M )) Ci

(i = 1, . . . , r),

Zr+1(M )

(i = 1, . . . , r),

E0 (Zr+1(M )) Cr+1

0 , respectively. As before, we put these represent A0 , ϕ 20 (A0 ), . . . , ϕ r0 (A0 ) and χ r+1 blocks together in an n × n matrix:

H(M )  0 E  := C1

··· ···

 X1 E 0  .. . Cr 0

Y1 .. . 0

··· .. . ···

0 .. .

Xr

  0 0 E .. , (Zr+1(M )). . Cr+1 Yr

The n × e columns of H(M ) are exactly the representatives of


201

0 A, A0 , ϕ(A), ϕ 0(A0 ), . . . , ϕ r (A), ϕ r0 (A0 ), χ r+1 (A00 ),

in this order. If M is in general position, then H(M ) is invertible. As before, consider the matrix H(M )−1M, which equals  E 0 ··· 0 0 0 a1 b1,r+2 c1,r+2 ··· b1,s c1,s a2 b 2,r+2 c1,r+2 ··· b1,s c1,s   0 E ··· 0 0 0 . . . ..  .. .. .. .. .. .. .. ..  . . . . . . . . . . . . . . . .    0 0 · · · E 0 0 a 2r−1 b 2r−1,r+2 c 2r−1,r+2 · · · b 2r−1,s c 2r−1,s    0 0 ··· 0 E 0 a b c ··· b c 2r

0

0

···

0

0

E

a 2r+1

2r,r+2

b 2r+1,r+2

2r,r+2

c 2r+1,r+2

2r,s

···

b 2r+1,s

2r,s

c 2r+1,s

0 (A00 ) implies Moreover, the property Er+1 = 8(A, ϕ 2 , . . . , ϕ r ) + χ r+1

a 2 = a 4 = · · · = a 2r = a 2r+1 = 0. Using the property W 0 := 8(A0 , ϕ 20 , . . . , ϕ r0 ) ∈ Er+2 , we may assume that W 0 is represented by the block  b  1,r+2 .   .. b 2r+1,r+2 and hence b1,r+2 = b 3,r+2 = · · · = b 2r+1,r+2 = 0. Furthermore, the matrix components of the map (4) can be calculated directly as c 2,r+2 b a c1,r+2 , D 2,r+2 , . . ., Z(M ) = D 1 a 3 c 3,r+2 b4,r+2 c4,r+2 b 2,r+2 c1,r+2 c 2,r+2 a1 ,D (14) D a 2r−3 c 2r−3,r+2 b 2r−2,r+2 c 2r−2,r+2 and

a1 b1,i b 2,r+2 c 2,i a1 b1,i ,D , . . ., D , 2i (M ) = D a 3 c 3,i b4,r+2 c4,i a 2r−1 b 2r−1,i b 2,r+2 c 2,r+2 −1 −1 , (c 2r+1,r+2 b 2r+1,i ), (c 2r+1,r+2 c 2r+1,i ) (15) D b 2r,r+2 c 2r,r+2

for i = r + 3, . . . , s, where D is the generalized double ratio defined by (5). Comparing this with the proof of Corollary 4.6 and applying the theorem of Procesi [P2] yields the following. Theorem 5.3. Let e, r, s be arbitrary positive integers such that s ≥ r + 2. Then the field of rational (GL (2r+1)e × GLs2e -invariants of (2r + 1)e × 2es matrices is generated by the functions Tr(σ1 · · · σk ), where each σl is either a component of the map Z in (14) or of one of the maps 2r+3 , . . . , 2s in (15) and k < 2e .

202

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Mathematisches Institut Eberhard-Karls-Universität Tübingen 72076 Tübingen Germany [email protected]