Lagrangian Grassmannians and spinor varieties in characteristic two

The vector space of symmetric matrices of size n has a natural map to a projective space of dimension 2^n-1 given by the principal minors. This map extends to the Lagrangian Grassmannian LG(n, 2n) and over the complex numbers the image is defined, as a set, by quartic equations. In case the characteristic of the field is two, it was observed that, for n = 3, 4, the image is defined by quadrics. In this paper we show that this is the case for any n and that moreover the image is a spinor variety of (n + 1)-dimensional maximal subspaces of a smooth quadric. Since some of the motivating examples are of interest in supergravity and in the black-hole/qubit correspondence, we conclude with a brief examination of some other cases related to integral Freudenthal triple systems over integral cubic Jordan algebras.


Introduction.
In the paper [HSL] the maximal commutative subgroups of the n-qubit Pauli group were studied. Such subgroups correspond to points in a Lagrangian Grassmannian LG(n, 2n) over the Galois field F 2 with two elements. A subset of this Grassmannian is parametrized by symmetric n × n matrices. The principal minor map π : S n := {symmetric n × n matrices} −→ PF 2 n 2 , which associates to a symmetric matrix with coefficients in F 2 its principal minors, extends to a map, again denoted by π, on all of LG(n, 2n): π : S n := LG(n, 2n) −→ Z n (⊂ PF 2 n 2 ) , where the image Z n of π is called the variety of principal minors of symmetric matrices.
Over the field of complex numbers, the variety Z n was studied in [HS] and in [O1] quartic equations which define Z n , as a set, were obtained. In case n = 3, Z n is defined by a unique quartic polynomial which is Cayley's hyperdeterminant. Returning to the case of the field F 2 , it was observed that the hyperdeterminant reduces to the square of a quadratic polynomial over this field and Z 3 is the quadric in PF 3 2 defined by this quadratic polynomial. Moreover, in [HSL] it was shown that for n = 4 the variety Z n is defined by ten quadrics in PF 16 2 . We will show that over any field of characteristic two, Z n is defined by quadrics for any n ≥ 3. Moreover, these quadrics define the (image of the) well-known spinor variety S n+1 associated to the group Spin(2n + 2): Over the complex numbers there is a natural embedding σ : S n+1 −→ PC 2 n , where C 2 n is (any) one of the two half-spin representations of Spin(2n + 2). Considering now a field of characteristic two, one obtains similarly an embedding σ : S n+1 −→ P 2 n −1 .
It is well-known that the image of σ (and of σ) is defined by quadrics. The spinor variety S n+1 parametrizes certain maximally isotropic subspaces of a smooth quadric. A subset of these subspaces is parametrized by alternating (n + 1) × (n + 1) matrices (since the characteristic is two, that means t A = −A (which is A!) and all diagonal coefficients of A should be zero). The maps σ and σ are given by the 2 n Pfaffians of the principal submatrices of A. The restriction of σ to these subspaces will again be denoted by the same symbol: σ : A n+1 := {alternating (n + 1) × (n + 1) matrices } −→ P 2 n −1 .
To show that Z n = σ(S n+1 ), we will define in Section 1.6 an explicit map α : S n −→ A n+1 , such that π(S n ) = σ(α(S n )) for all symmetric matrices S n ∈ S n with coefficients in a(ny) algebraically closed field of characteristic 2. The proof involves an 'induction on n' argument and the verification of a quadratic relation between the determinant of a symmetric matrix and certain of its principal minors, see Proposition 2.2. The map α extends to a map α : LG(n, 2n) −→ S n+1 .
To complete the picture, we discuss in Section 4 a classical map, over fields of characteristic two, β : S n+1 −→ LG(n, 2n), βα = F LG(n,2n) , αβ = F Sn+1 , where F is the Frobenius map, which is induced by the map (. . . : x i : . . .) → (. . . : x 2 i : . . .) on the projective spaces. This points to an 'exceptional' isogeny of linear algebraic groups in characteristic two as the 'reason' for these results, see Remark 4.4.
Since both the Cayley hyperdeterminant and S 6 appear in the context of Freudenthal triple systems and four-dimensional Maxwell-Einstein supergravity on four space-time dimensions (as well as in [Ho, Table 3]), we add a brief discussion on some characteristic two aspects of that topic.
1 The maps.
1.1 The fields. Even if our motivation comes from algebra and geometry over the Galois field F 2 with two elements, we will consider the case of an algebraically closed field k of characteristic two. In such a field 2 = 0 (and −1 = +1), in particular the finite 'binary' field F 2 = Z/2Z is contained in k, but k will have infinitely many elements and one can do algebraic geometry over such a field as well.
1.2 Principal minors of symmetric matrices. We recall the basics of the principal minors of a symmetric matrix. Let be a symmetric n × n matrix. For a subset I := {i 1 . . . , i k } of {1, . . . , n} with 1 ≤ i 1 < . . . < i k ≤ n, the principal minor defined by I is the determinant of the submatrix of S n with coefficients (S n ) ij with i, j ∈ I. This principal minor will be denoted by S n,I and if I is the empty set we put S n,∅ = 1. For example, The principal minor map π : S n → P 2 n −1 is defined by the 2 n = n 0 + n 1 + . . . + n m + . . . + n n principal minors of the m × m principal submatrices with 0 ≤ m ≤ n.
Then A corresponds to a 2-form where the e i are the standard basis of R N . In case N is even, one defines a homogeneous polynomial Pf(A) ∈ Z[. . . , y ij , . . .] 1≤i<j≤N of degree N/2 by considering the N/2-th exterior power of σ A : In case N is odd, we simply put Pf(A) = 0. For any field k there is a natural homomorphism of rings Z → k defined by sending 1 ∈ Z to 1 ∈ k and this extends to a homomorphism of rings Z[. . . , y ij , . It is not hard to verify the following formula for the Pfaffian of an alternating N × N matrix A with coefficients y ij : where A1ĵ is the (N − 2) × (N − 2) submatrix of A where the first and j-th row and column of A are deleted. In case char(k) = 2 and n is a fixed integer with 1 ≤ n ≤ N one similarly has the following formula (we omit a sign since char(k) = 2 and notice that y jj = 0): For any subsetĨ ⊂ {1, . . . , N } with an even number of elements we consider the 'principal' submatrix of A with coefficients (A N ) ab and a, b ∈Ĩ. These matrices are again alternating and thus we can consider their Pfaffians, which we denote by A N,Ĩ and we put A N,∅ := 1. For example: A N,{i,j,k,l} = y ij y kl − y ik y jl + y il y jl .
The Pfaffian map σ : A N → P 2 N −1 −1 is defined by the Pfaffians of the m × m principal submatrices, with m even and 0 ≤ m ≤ N . Since e i ∧ e j and e k ∧ e l commute in the exterior algebra ∧ * k N , one easily verifies that, with A = A N , where the sum is over the ordered subsetsĨ = {i 1 , . . . , i 2k } ⊂ {1, . . . , N } with an even number of elements and eĨ = e i1 ∧ . . . ∧ e i 2k , since the k! in the definition of Pf(AĨ ) cancels with the 1 k! in the exponential function. Using commutativity as well as (e i ∧ e j ) ∧2 = (e i ∧ e j ) ∧ (e i ∧ e j ) = 0, we also have: and thus: a formula which works over any field, also of finite characteristic. The Pfaffian map now appears as a natural map from A N into P ∧ even k N . In particular, the image of σ : A 4 → P 7 is defined by the degree two polynomial z 000 z 111 + z 001 z 110 + z 010 z 101 + z 100 z 011 and thus, comparing with Section 1.3, the polynomials defining the images of σ (for n = 4) and π (for n = 3) are the same (and this holds over any field of characteristic two).
1.6 A map from symmetric to antisymmetric matrices. In Section 1.5 we observed that, over a field with characteristic two, the maps π and σ, with domains S 3 and A 4 respectively, have images that are defined by the same quadratic polynomial. Now we define a map α : S n → A n+1 which will be shown to have the property: π(S n ) = σ(α(S n )) for any n.
With the notation from Sections 1.2, 1.4, we define a (non-linear) map Notice that we assume the field to have characteristic two, soS n is alternating (in fact, (S n ) ii = 0 for all i and (S n ) ij = −(S n ) ji = (S n ) ji ). For example, x 11 x 12 x 13 x 12 x 22 x 33 x 11 x 22 + x 2 12 x 11 x 33 + x 2 13 x 11 x 11 x 22 + x 2 12 0 x 22 x 33 + x 2 23 x 22 x 11 x 33 + x 2 13 x 22 x 33 + x 2 23 0 x 33 x 11 x 22 Finally we define how the coordinate functions of π and σ correspond: for any subset I ⊂ {1, . . . , n} we define a subsetĨ ⊂ {1, . . . , n + 1} with an even number of elements as follows: We will prove the following theorem in Section 2: 1.7 Theorem. Let π : S n → P 2 n −1 be the principal minor map with coordinate functions S n,I as in Section 1.2 and let σ : A n+1 → P 2 n −1 be the Pfaffian map with coordinate functions A n+1,Ĩ as in Section 1.4 and where I andĨ correspond as above. Let α : S n → A n+1 be defined as in Section 1.6.
Then we have, over any field of characteristic two : In fact, S n,I =S n,Ĩ for all S n ∈ S n and all subsets I of {1, . . . , n}.
1.8 Examples. We give some examples of the identity S n,I =S n,Ĩ . Obviously S n,∅ = 1 =S n,∅ . In case I = {i} one hasĨ = {i, n + 1} and indeed S n,{i} = x ii =S n,{1,n+1} . In case I = {i, j} one hasĨ = I and we do have the identity: which holds since: Notice that these examples show that for n = 3 we have S 3,I =S 3,Ĩ for all subsets I of {1, 2, 3}. Thus we verified Theorem 1.7 for n = 3, this will be the starting point for an induction argument.
2 The proof of Theorem 1.7.
Proof. We need to show that S n,I =S n,Ĩ for any n and any I ⊂ {1, . . . , n}. We proceed by induction on n, and we already verified the equalities for all I in the case n = 3. So we assume that S n,I =S n,Ĩ holds for all I ⊂ {1, . . . , n} and we must prove that S n+1,J =S n+1,J for all subsets J ⊂ {1, . . . , n + 1}. In case ♯J < n+1, after a permutation of the indices, we may assume that J = {1, 2, . . . , k} ⊂ {1, . . . , n}, and then S n+1,J =S n+1,J follows from the induction hypothesis. To deal with the remaining case J = {1, . . . , n+1} we distinguish the cases n + 1 odd and n + 1 even.
In case n + 1 is odd,J = {1, . . . , n + 1, n + 2} and we must show that S n+1,J =S n+1,J , that is det(S n+1 ) = Pf(S n+1 ). It is more convenient to change the integer n to n − 1 and then we must show det(S n ) = Pf(S n ) for n odd. Using the formula for computing the Pfaffian given in Section 1.4 (with N = n + 1) we have The principal submatrixS n,n ofS n obtained by deleting the n-th row and column, is an alternating n×n matrix where the coefficients x in no longer appear and which is exactlyS n−1 , soS n,n =S n−1 . For all k ∈ {1, . . . , n − 1} the Pfaffian of the (n − 1) × (n − 1) alternating matrixS n−1,k obtained by deleting the k-th row and column ofS n−1 isS n−1,Ĩ whereĨ = {1, . . . ,k, . . . , n}. By induction we know that this Pfaffian is det(S n−1,I ) where I = {1, . . . ,k, . . . , n − 1} in case k < n, which is also det(S n−1,k ). In case k = n, we have (S n ) n,n = 0 and we already omitted this term. Finally if k = n + 1 we haveS n,n, n+1 =S n−1,Ĩ whereĨ = {1, . . . , n − 1} and thus, by induction, Pf(S n,n, n+1 ) = det(S n−1 ). Thus we can rewrite the Pfaffian ofS n in terms of principal minors of S n−1 : and the equality det(S n ) = Pf(S n ) for n odd follows from Proposition 2.2.2 below. In case n + 1 is even, J = {1, . . . , n + 1} =J and we must show that S n+1,J =S n+1,J , that is det(S n+1 ) = Pf(S n+1, n+2 ). Again we prefer to change the integer n to n − 1, so we must show that for n even we have det(S n ) = Pf(S n, n+1 ). We have the following expansion of the Pfaffian of the alternating n × n matrixS n, n+1 : Notice thatS n,k,n, n+1 =S n−1,k,n and by induction we may assume that Pf(S n−1,k,n ) = det(S n−1,k ) since if n is even, then I := {1, . . . ,k, . . . , n − 1} =Ĩ. Finally we notice that S n−1,k = S n,k,n . Thus the equality det(S n ) = Pf(S n, n+1 ) for n even follows from Proposition 2.2.1 below. So Theorem 1.7 now follows from Proposition 2.2 below.
2.2 Proposition. Let k be a field of characteristic two and let S n = (x ij ) be a symmetric n × n matrix. Then we have the following relation between principal minors of S n : 1. in case n is even, where det(S n,î,ĵ ) is the principal minor S n,I with I the subset of {1, . . . , n} with only i, j omitted and S n−1 = S n,n is the submatrix of S n where the last row and column are omitted.

The determinant of a symmetric matrix. The determinant of an
where Σ n is the symmetric group on {1, . . . , n}. As det(A) = det( t A), under the substitution a ij := a ji the monomials of the determinant are either fixed or permuted in pairs. A fixed term may contain any a ii 's and if a ij occurs, so does a ji . In a field of characteristic two, one has +1 = −1 and x + x = 0, so in a determinant of a symmetric matrix over such a field the paired monomials will cancel and only the fixed monomials appear, all with coefficient 1. If a ij , with i = j, occurs in a fixed term, then since a ij = a ji , the term contains a 2 ij . Up to a simultaneous permutation of the rows and columns (to preserve the symmetry) any term in the determinant of the symmetric matrix S n is thus of the form: n−1,n k = 0, 1, . . . , n .
2.4 Proof of Proposition 2.2. The right hand sides of the two formulas in 2.2 are invariant under simultaneous permutations of row and columns which fix the last row and column. This reduces the verification of the formula to the following terms: Notice that the t k appearing on the left hand side are those for which n and k have the same parity. Similarly, the t ′ k on the left are those for which n and k have different parity. On the right hand side, each term in (x ii x nn + x 2 in ) det(S n, i,n ) and also in x nn det(S n−1 ) is a t k or a t ′ k up to simultaneous permutation of rows and columns. So we only need to verify that each term of type t k occurs an odd number of times in the summands on the right hand sides of 2.2.
The terms t k all have the variable x n−1,n . In the matrices S n,î,n (i = 1, . . . , n − 1) and S n−1 appearing in the two formulas in 2.2 we omit the n-th row and column, so they don't have the variable x n−1,n . Only x n−1,n−1 x nn + x 2 n−1n has this variable. Each t k , k = 0, . . . n − 2, thus occurs at most once in the expansion of the right hand side. It is also not hard to see that each t k actually occurs in x 2 n−1n · det(S n, n−1,n ), provided k has the same parity as n.
Now consider the terms t ′ k . We notice first of all that t ′ n−1 = x 11 · . . . · x nn occurs in all terms on the right hand side of each of the two formulas in 2.2 and since the two right hand sides each have an odd number of terms, it survives.
Next we consider t ′ n−3 = x 11 · . . . · x n−3,n−3 x 2 n−2,n−1 x n,n . Considering x 2 n−2,n−1 , it obviously does not occur in the two terms However, t ′ n−3 does appear in all other summands of each of the two right hand sides 2.2. Thus t ′ n−3 appears in an odd number of summand and hence it appears on the right hand side. More generally, t ′ n−2k does not appear in the 2k summands (x n−i,n−i x nn + x 2 n−i,n ) det(S n, n−i,n ) for i = 1, . . . , 2k, but it appears in all other summands. Hence t ′ n−2k appears in an odd number of summands and hence it appears on the right hand side. This concludes the proof of Proposition 2.2.
3.1 Global aspects. We recall that the spaces of symmetric and antisymmetric matrices have a natural interpretation as open subsets of certain Grassmannians, like the spinor varieties, and that the principal minor map π and the Pfaffian map σ extend to these Grassmannians. We also discuss the actions of some groups on these Grassmannians. In the final section we recall that the image of the spinor variety is defined by quadrics.

The Lagrangian
Grassmannian. Let V be a vector space over a field k and let be a symplectic form, that is, an alternating, non-degenerate, bilinear form (so for any x ∈ V , e(x, x) = 0 and if x = 0, there is a y ∈ V with e(x, y) = 0). Then V has a symplectic basis f 1 , . . . , f 2n that is e(f i , f j+n ) = −e(f j+n , f i ) = δ ij (Kronecker's delta) for 1 ≤ i, j ≤ n and all other e(f i , f j ) are zero. So if I denotes the n × n identity matrix, then A (linear) subspace W ⊂ V is called isotropic if e(w, w ′ ) = 0 for all w, w ′ ∈ W and W is called Lagrangian if it is isotropic and dim W = n, the maximal possible. Choosing a basis w 1 , . . . , w n of W , let M W be the 2n × n matrix whose columns are the w i . Then W = im(M W : k n → k 2n ) and W is isotropic iff In particular, the subspace W 0 :=< f 1 , . . . , f n > is Lagrangian and M W0 has blocks A = I and B = 0. More generally, given a symmetric n × n matrix X, the subspace W X spanned by the columns of the matrix M with blocks A = I and B = X is Lagrangian. .
The Lagrangian subspaces of k 2n are parametrized by the Lagrangian Grassmannian LG(n, 2n), an algebraic subvariety of dimension n(n + 1)/2 of the Grassmannian Gr(n, 2n) of all n-dimensional subspaces of k 2n . If e is the standard symplectic form on k 2n and thus the f i are a symplectic basis of k 2n , then we define e * := n i=1 f i ∧f i+n ∈ ∧ 2 k 2n . Then one can show that W is a Lagrangian subspace iff (∧ n W )∧e * = 0 ∈ ∧ n+2 k 2n . Thus if we denote by (∧ n k 2n ) 0 the kernel of the wedge product with e * : (∧ n k 2n ) 0 := ker(∧ n k 2n −→ ∧ n+2 k 2n , ω −→ ω ∧ e * ) , then we find
In particular, if n = 3 the dimension is 20 − 6 = 14, see [IR] for a study of this case.

3.4
The principal minor map. The principal minor map extends to a map, again denoted by π, where J runs over the 2 n special subsets J ⊂ {1, . . . , 2n} with ♯J = n, where, for every i ∈ {1, . . . , n}, J contains either i or n + i. In case W is the image of M W and M W has blocks I and X ∈ S n , then these p J (W ) are easily seen to be the principal minors of X. Thus π is a projection of LG(n, 2n) ⊂ P(∧ n k 2n ) 0 into P 2 n −1 and it is not hard to verify that π is a regular map (base point free) on LG(n, 2n). The closure Z n of π(S n ) is thus the projective variety π(LG(n, 2n)). We now show that the morphism π : LG(n, 2n) → Z n has degree 2 n−1 , at least if the characteristic of k is not two (and if it is two, the map is probably purely inseparable of degree 2 n−1 ). (In the lemma below, LG(n, 2n)/G n is not isomorphic to Z n for n > 3 since there are invariant monomials in the x ij on S n ⊂ LG(n, 2n) which are not contained in the ring of principal minors.) 3.5 Lemma. The principal minor map π : LG(n, 2n) → Z n (⊂ P 2 n −1 ) has degree 2 n−1 over a field of characteristic different from two. This map factors over a quotient of LG(n, 2n) by a group G n ∼ = (Z/2Z) n−1 .
Proof. Any diagonal matrix D = diagonal(t 1 , . . . , t n , t −1 1 , . . . , t −1 n ) with t i = 0 fixes the symplectic form e and thus maps LG(n, 2n) into itself by W → DW , equivalently, M W → DM W . Let D 1 := diagonal(t 1 , . . . , t n ), and notice that DM W and DM W D −1 1 map k n to the same subspace DW in k 2n . For M W with blocks I, X, the matrix DM W D −1 1 has blocks I, D −1 1 XD −1 1 , so we see that D maps the image of S n in LG(n, 2n) into itself and acts as D : X → D −1 1 XD −1 1 . In case all t i ∈ {1, −1}, we have D −1 1 = D 1 and we write more suggestively D : X → D 1 XD −1 1 , the conjugation by D 1 . Any principal submatrix of X is then also conjugated by a submatrix of D 1 , and hence the principal minors of X and those of D 1 XD −1 1 are the same. So the fiber of π over π(X) contains all the D 1 XD −1 1 where D 1 has coefficients ±1. Obviously D 1 = −I acts trivially and thus we have an action of the group G n := (Z/2Z) n−1 on LG(n, 2n) and π factors over LG(n, 2n)/G n . The Since the x ii , x ii x jj − x 2 ij are principal minors of X, we can recover the x ij from π(W X ), except for the signs of the x ij with i = j. However, the principal minors S n,{i,j,k} (see Section 1.2) show that once, for a fixed i, all the x il are non-zero and the signs of all these x il are fixed, then the signs of all x jk are fixed. Therefore the fiber over π(X), for general X ∈ S n , consists of exactly 2 n−1 elements that are an orbit of G n . This implies that π has degree 2 n−1 and that π factors over LG(n, 2n)/G n .

The spinor varieties. A quadratic form on a vector space V over a field k is a map
where a ∈ k and e is a bilinear form and x, y ∈ V . We consider the quadratic form q on V = k 2n defined by A (linear) subspace W ⊂ V is called an isotropic subspace of q if q(w) = 0 for all w ∈ W and it is a maximally isotropic subspace of q if moreover dim W = n, the maximum possible. Choosing a basis w 1 , . . . , w n of W , let M W be the 2n × n matrix whose columns are the w i . Then W = im(M W : k n → k 2n ). The subspace W is maximally isotropic for q iff q(w i ) = 0, q(w i + w j ) = 0, 1 ≤ i, j ≤ n, in fact, if q(w i ) = 0 and also 0 = q(w i + w j ) = e(w i , w j ) for all i, j, then from we see that W is maximally isotropic. In case char(k) = 2 this can also be checked using the symmetric matrix of e: and notice that q(w i ) = ( t AB + t BA) ii and q(w i + w j ) = ( t AB + t BA) ij . The subspace W 0 :=< f 1 , . . . , f n > is thus maximally isotropic for q. More generally, given an antisymmetric n × n matrix Y , the subspace W Y spanned by the columns of the matrix M with blocks A = I and B = Y is Lagrangian, so This holds over any field, since q(w i ) = y ii and q(w i + w j ) = y ii + y jj + y ij + y ji and thus W Y is maximally isotropic for q iff the diagonal coefficients of Y are zero and y ij + y ji = 0 iff Y is alternating. There are two n(n−1)/2-dimensional families of maximally isotropic subspaces of q. They parametrized by the spinor varieties S n,ev and S n,odd which are isomorphic and we denote the one containing W 0 by S n . For spinor varieties see [Ch], [P,Section 11.7] and the references given in [RS,Section 6.0].

3.7
The image of the Pfaffian map. The Pfaffian map on A n from Section 1.4 extends to an embedding of the spinor variety σ : S n −→ P 2 n−1 −1 .
The spinor variety S n is the homogeneous variety G/P , with G = Spin(2n) and the image of σ consists of the pure spinors (for any one of the two half spin representations of G), as in [P, 11.7.2], σ(S n ) is also the Gorbit of the highest weight vector in the projectivization of the half spin representation. Under certain natural identifications, the Lie algebra of the Spin group is identified with a subspace of the Clifford algebra C(q) of q and a maximally isotropic subspace W of q defines a subalgebra ∧ * W ⊂ C(q). In case e 1 , . . . , e n is a basis of W , the element exp(y ij e i ∧ e j ) = (1 + y ij e i ∧ e j ) introduced in Section 1.4 is actually an element of the Spin group and from this one can deduce that the orbit of the highest weight vector is indeed locally parametrized by the Pfaffian map.
In general, the orbit under a semisimple simply connected algebraic group G (defined over an algebraically closed field of arbitrary characteristic) of a highest weight vector in an irreducible minuscule representation of G is the intersection of quadrics, see [Se]. This implies that the image of σ is an intersection of quadrics. The number of quadrics can also be determined, it is dim I 2 := 2 n−1 + 1 2 − 1 2 2n n , where k[. . . , z I , .
. .] is the homogeneous coordinate ring of P 2 n−1 −1 , in fact, [Se] shows that dim I 2 does not depend on the characteristic of the field and over the complex numbers one can use for example (the proof of) [vG,Theorem 2]). So for n = 4, 5, 6 we find 36 − 35 = 1, 136 − 126 = 10, 528 − 462 = 66 quadrics respectively. See also the end of section [P, 11.7.2] for the quadratic relations between Pfaffians, [SV] for explicit methods to find the quadratic equations of σ(S n ) and [RS,Section 6] for a study of the case n = 5.
3.8 Proposition. Let π : S n → P 2 n −1 be the principal minor map over an algebraically closed field of characteristic two. Then the closure Z n of the image of π is σ(S n ) and in particular Z n is an intersection of quadrics.
In Section 3.7 we recalled that σ(S n+1 ) is defined by quadrics, hence also Z n is defined by quadrics.
4.1 From antisymmetric to symmetric matrices. We work over a field of characteristic two. In Section 1.6 we defined α : S n → A n+1 in such a way that the principal minors of S n where the Pfaffians of α(S n ), this condition determined the map α. Now we consider a map β : A n+1 → S n , which is defined in terms of a well-known map from S n+1 → LG(n, 2n), which we will also denote by β. The maps α and β are not mutual inverses, instead their compositions are purely inseparable maps, given by squaring all coefficients in the matrix.
Since the field has characteristic two, these maps are injective and if the field is algebraically closed (or more generally, if it is perfect) then these maps are bijections. Let A n+1 = (y ij ) ∈ A n+1 be an alternating (n + 1) × (n + 1) matrix (so y ii = 0 and y ij = y ji ) and define For example, y 12 + y 14 y 24 y 13 + y 14 y 34 y 12 + y 14 y 24 y 2 24 y 23 + y 24 y 34 y 13 + y 14 y 34 y 23 + y 24 y 34 y 2 34   .
It is not hard to verify that for all i, j = 1 . . . , n and all k, l = 1, . . . , n + 1. Thus the maps βα : S n → S n and αβ : A n+1 → A n+1 are the (coordinate wise) Frobenius maps on the respective vector spaces of matrices: 4.2 The geometry of β. We explain the geometry behind the map β, we denote the field of characteristic two by k. In Section 3.6 we considered an embedding A n+1 ֒→ S n+1 , where S n+1 parametrizes certain maximally isotropic subspaces for the quadratic form q(y) = n+1 i=1 y i y n+1+i on k 2n+2 . We define a hyperplane H : y n+1 + y 2n+2 = 0 (⊂ k 2n+2 ) .
The intersection H ∩ (q = 0) can be identified with the quadric in k 2n+1 defined by q ′ , simply by mapping z = (z 1 , . . . , z 2n+1 ) → y = (z 1 , . . . , z 2n+1 , z n+1 ) ∈ H. A linear subspace contained in q ′ = 0 has dimension at most n and there is a unique family of such subspaces. If W ⊂ (q = 0) is a maximal isotropic subspace for q, so dim W = n + 1, then W ′ := W ∩ H is a subspace of q ′ = 0 of dimension ≥ n + 1 − 1 = n and we conclude that W ′ must have dimension n, so W ′ is maximally isotropic in q ′ = 0. This sets up a bijection between S n+1 and the spinor variety that parametrizes the maximally isotropic subspaces for q ′ .
Proof. Given Y ∈ A n+1 , let W Y ⊂ k 2n+2 be the subspace spanned by the columns of the (2n + 2) × (n + 1) matrix M with blocks I and Y . The intersection W ′ Y := H ∩ W Y is spanned by the n vectors c i + y i,n+1 c n+1 , i = 1, . . . , n, where c i is the i-th column of M , in fact the n + 1 and 2n + 2 coefficients of c i + y i,n+1 c n+1 are 0 + y i,n+1 · 1 and y i,n+1 + y i,n+1 · 0 respectively, and their sum is indeed zero, showing that these vectors do lie in H ∩ W Y . Next we project these vectors to k 2n , so we omit the n + 1-st coefficients, their span is then H ∩ W Y . The image vectors are the columns of the 2n × n matrix with blocks I and Y , which proves that β induces Y → Y .
4.4 Remark. The underlying reason for the results we obtained thus seems to be the isogeny of the linear algebraic groups SO(2n + 1) → Sp(2n) (of type B n and C n respectively) over a field of characteristic two, cf. [St,4.11], [Mi], [CGP,7.1,Remark 7.1.6]. The description of the isogeny leads directly to the map β : S n+1 → LG(n, 2n). It might be interesting to use this isogeny to relate the spaces of global sections of line bundles on S n+1 and LG(n, 2n) and thus to show more intrinsically how principal minors and pfaffians are related in characteristic two. 5 Freudenthal triple systems.
It should be noticed that one usually excludes fields of characteristic 2 and 3 in these subjects. However, e.g. in [BDDR] integral Freudenthal triple systems and integral cubic Jordan algebras have been studied, and these can be reduced modulo two. Below we will also consider an approach to these subjects through the algebraic geometry of tangential varieties of certain homogeneous spaces.
5.2 Groups of type E 7 . A group of type E 7 is the subgroup, which we denote by G 4 , of GL(R), where R is a finite-dimensional vector space over a field k, which preserves a non-degenerate alternating form e and a homogeneous quartic polynomial q (cf. [B], [K]). It turns out that given some additional conditions, including a compatibility between e and q and char(k) = 2, 3, as well as the condition that − 1 2 q(f, f, f, f ) is a non-zero square for some f ∈ R (such triples (R, q, e) are called reduced, [B, p.90]), the vector space R decomposes as where J is a (cubic) Jordan algebra [JVNW], in [K,Section 3.1] R is denoted by M (J). The Jordan algebras of interest for us are those given by the 3 × 3 matrices of the form where C is a composition algebra with involution x →x. and norm n : C → k. The best known examples are k = R and C = R, C, H, O, where H are the quaternions and O are the octonions. For simplicity we will also consider the corresponding split algebras as in [K,Section 2.1] here. Notice that the dimension of J as a k-vector space is 3q + 3 where q := dim C and dim R = 2 + 2(3q + 3) = 6q + 8. Over the complex numbers (the split case), the group G 4 will be SL(2, C) 3 , Sp(6, C), SL(6, C), Spin(12, C) and E 7 (C) for q = 0, 1, 2, 4, 8, respectively, cf. [Ho, Table 3]. The G 4 -orbit Y q of the highest weight vector in PR = P 6q+7 is a complex projective algebraic variety of dimension 3q + 3, it is the unique closed orbit of G 4 in PR. The tangential variety X q of Y q (see also Section 5.8) has dimension 6q + 6 and is defined by the quartic polynomial q (cf. [FGK], [LM], [Ho]).

5.3
The alternating form and the quartic. The algebra J comes with a norm N : J → k, which is homogeneous of degree three, and which generalizes the determinant of a 3 × 3 matrix: N (A) = abc − axx − byȳ − czz + (xy)z +z(ȳx) . Finally there is 'sharp' operation on J, similar to the adjoint of a matrix, We will write elements of R as four tuples (a, A, B, b) with a, b ∈ k and A, B ∈ J but often a matrix notation is used (cf. [K, (29)]). With these definitions (cf. [K,Section 2.4 so it becomes the square of a quadratic polynomial. This may be particularly relevant when considering integral Freudenthal triple systems in characteristic 2, since in this case the so-called Freudenthal duality is always defined, albeit becoming simply an antinvolutive electric-magnetic symplectic duality transformation ( [BDDR] and [FKM]). Since (A, B) is symmetric and bilinear in A, B the bilinear form associated to (A, B) − ab, defined as Q(x + y) − Q(x) − Q(y), is: hence (notice that '+ ′ = '− ′ ) the associated bilinear form is now the alternating form e. In general, the associated bilinear form e of a quadratic form over a field of characteristic two is alternating (that is e(x, x) = 0) which follows easily by putting x = y in Q(x + y) − Q(x) − Q(y) = e(x, y).
In particular, the group fixing both the alternating form and the quadric is just the orthogonal group fixing the quadric, see also [K,Remark 20]. The case q = 1 presents some extra features since in that case the quadric and the alternating form are degenerate, see Example 5.6. 5.6 Example: q = 1. We consider the constructions from Section 5.3 for the case that k = C = R, with x = x and n(x) = x for x ∈ C. Then q = dim k C = 1 and dim J = 6, dim R = 14 and R = (∧ 3 k 6 ) 0 , which is an irreducible Sp(6, R)-representation, cf. Section 3.3. In that case J is just the six-dimensional R-algebra of 3 × 3 symmetric matrices, N (A) = det(A) and A ♯ is the adjoint matrix of A and we write A = (a ij ), . . ., D = (d ij ) where A, . . . , D are 3 × 3 symmetric matrices. Then e (((a, A, B, b), (c, C, D, d)) = a 11 d 11 −b 11 c 11 +a 22 d 22 −b 22 c 22 +a 33 d 33 −b 33 c 33 +2 a 12 d 12 −b 12 c 12 +a 13 d 13 −b 13 c 13 +a 23 d 23 −b 23 c 23 ) +ad − bc, notice the factors 2 which appear. The quartic q ′ (x) = a 2 11 b 2 11 + 4a 11 a 22 a 33 t 1 + . . . + t 2 1 t 2 2 has 44 terms. Reducing modulo two one finds: q ′ ≡ Q 2 , Q := a 11 b 11 + a 22 b 22 + a 33 b 33 + t 1 t 2 , and Q = 0 is a singular quadric in P 13 (notice that the 6 = 3 + 3 variables a ij , b ij with i < j do not appear).
The associated bilinear form of Q is the reduction of e mod 2, and this is a degenerate alternating form. Notice that while [K] discusses the next three cases, q = 2, 4, 8, the case q = 1 is avoided.

5.7
The cases q = 2, 4, 8. The cases q = 2, 4, 8 do not seem to present special features. In case q = 8 (see also [W]), we use the expression for the quartic invariant that we found in [Fr]. Let X := (x ij ), Y := (y ij ) be alternating 8×8 matrices over the real or complex numbers. We view the x ij , y ij as coordinates on a 28+28 = 56 dimensional vector space and we define a symplectic form on this vector space by requiring that the x ij , y ij are the coordinates on a symplectic basis. The quartic invariant of E 7 is defined as J := Pf(X) + Pf(Y ) − 1 4 Tr(XY XY ) + 1 16 (Tr(XY )) 2 .
The degree four polynomial J has 1036 terms and coefficients in {±1, ± 1 2 , − 1 4 }. To find a reduction mod 2 we simply multiply J by 4 and then reduce mod 2 to obtain a quartic J which has only 28 terms: J = x 2 12 y 2 12 + x 2 13 y 2 13 + . . . + x 2 78 y 2 78 = 1≤i<j≤28 x 2 ij y 2 ij which is the square of the quadratic polynomial Q = i<j x ij y ij . Notice that the alternating form defined by Q is indeed the symplectic form defined above.
5.8 Tangential varieties. The representation of the the group G 4 on PR = P 6q+7 as a unique closed orbit Y q of dimension 3q + 3 and one recovers the zero locus of the quartic invariant as the 6q + 6-dimensional tangential variety X q of Y q . This allows one to determine the equation of X q over a field of characteristic two. In the case q = 1 we again find the singular quadric from Section 5.6 and in the cases q = 0, 2, 4 where we did the computations, we again find a smooth quadric as before. We didn't attempt to compute the q = 8 case. Given a subvariety Z of P N , its tangential variety, often denoted by τ (Z), is the union of all of its projective tangent spaces, equivalently, it is the union of all embedded tangent lines to Z, see [Z]. If locally we have a parametrization φ : k n −→ Z, φ(x) = (f 0 (x) : . . . : f N (x)) for certain functions on k n , then the tangent spaces to the image of φ are locally parametrized bỹ φ : k n × k n −→ P N , (x, y) −→ (. . . : f k (x) + n j=1 ∂f k ∂xj (x)y j : . . .) .
In particular, given an explicit local parametrization of Z with dense image, one can compute the homogeneous polynomials vanishing on the tangential variety of Z. For references and some recent results, mostly over the complex numbers, see [O2]. As we will see in the next section, over a field of characteristic two the tangential varieties behave rather differently from other characteristics (in all our examples we find quadrics instead of quartics). Although 'bad' behaviour of tangential varieties in characteristic two is not surprising, we are not aware of a good explanation for the results we found in the cases we considered.

5.9
The various cases. In this section we assume that the characteristic of the field we work over is two.
In case q = 0, we verified that the quadric we found in Section 5.4 is the tangential variety of the Segre 3-fold Y 0 , the image of P 1 × P 1 × P 1 in P 7 .
For q = 1, one finds by direct computation that the image Y 1 of LG(3, 6) under the Plücker map spans a P 13 ⊂ P 19 . The tangential variety X 1 of LG(3, 6) in this P 13 is a singular quadric of rank 8. The singular locus of the quadric is a P 5 ⊂ P 13 . Notice that this P 5 is mapped into itself under the action of Sp(6) on LG(3, 6) and P 13 .
For q = 2, the tangential variety of the Plücker embedded Y 2 = Gr(3, 6) ⊂ P 19 is a smooth quadric. Also for q = 4, we checked that the tangential variety of the 15-dimensional Y 4 = S 6 , Pfaffian embedded in P 31 , is a smooth quadric (cf. [Ma] for such aspects of the geometry of spinor varieties).
In case q = 8, the variety Y 8 of dimension 27 in P 55 is known as the Freudenthal variety [MM]. Its tangential variety X 8 , over the complex numbers, is the quartic hypersurface defined by J = 0 in Section 5.7. We haven't computed what happens over a field of characteristic two since the only parametrization of Y 8 that we know of is rather cumbersome.