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 2ⁿ −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 charac...
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| Цитувати: | Lagrangian Grassmannians and Spinor Varieties in Characteristic Two / B. van Geemen, A. Marrani // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 41 назв. — англ. |
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| author | van Geemen, B. Marrani, A. |
| author_facet | van Geemen, B. Marrani, A. |
| citation_txt | Lagrangian Grassmannians and Spinor Varieties in Characteristic Two / B. van Geemen, A. Marrani // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 41 назв. — англ. |
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| description | The vector space of symmetric matrices of size n has a natural map to a projective space of dimension 2ⁿ −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 the spinor variety associated to Spin(2n+1). 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 other cases related to integral Freudenthal triple systems over integral cubic Jordan algebras.
|
| first_indexed | 2025-12-07T21:24:50Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 064, 22 pages
Lagrangian Grassmannians and Spinor Varieties
in Characteristic Two
Bert VAN GEEMEN † and Alessio MARRANI ‡§
† Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133 Milano, Italy
E-mail: lambertus.vangeemen@unimi.it
‡ Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi,
Via Panisperna 89A, I-00184, Roma, Italy
§ Dipartimento di Fisica e Astronomia Galileo Galilei, Università di Padova,
and INFN, sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy
E-mail: alessio.marrani@pd.infn.it
Received March 08, 2019, in final form August 21, 2019; Published online August 27, 2019
https://doi.org/10.3842/SIGMA.2019.064
Abstract. The vector space of symmetric matrices of size n has a natural map to a pro-
jective space of dimension 2n − 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 the spinor variety associated to Spin(2n+ 1).
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 other cases related to
integral Freudenthal triple systems over integral cubic Jordan algebras.
Key words: Lagrangian Grassmannian; spinor variety; characteristic two; Freudenthal triple
system
2010 Mathematics Subject Classification: 14M17; 20G15; 51E25
1 Introduction
In the paper [21] 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 F2 with two elements. A subset of this Grassmannian is parametrized by symmetric n× n
matrices. The principal minor map
π : Sn := {symmetric n× n matrices} −→ PF2n
2 ,
which associates to a symmetric matrix with coefficients in F2 its principal minors, extends to
a map, again denoted by π, on all of LG(n, 2n):
π : LG(n, 2n) −→ Zn
(
⊂PF2n
2
)
,
where the image Zn of π is called the variety of principal minors of symmetric matrices.
Over the field of complex numbers, the variety Zn was studied in [19]. In [33] quartic equations
which define Zn, as a set, were obtained. In case n = 3, Zn is defined by a unique quartic
polynomial which is Cayley’s hyperdeterminant.
Returning to the case of the field F2, it was observed that the hyperdeterminant reduces to
the square of a quadratic polynomial over this field and Z3 is the quadric in PF3
2 defined by this
mailto:lambertus.vangeemen@unimi.it
alessio.marrani@pd.infn.it
https://doi.org/10.3842/SIGMA.2019.064
2 B. van Geemen and A. Marrani
quadratic polynomial. Moreover, in [21] it was shown that for n = 4 the variety Zn is defined
by ten quadrics in PF16
2 .
We will show that over any field of characteristic two, Zn is defined by quadrics for any n ≥ 3.
Moreover, these quadrics define the (image of the) well-known spinor variety Sn+1 associated to
the group Spin(2n+ 1):
Zn ∼= Sn+1
(
⊂P2n−1
)
.
Over the complex numbers there is a natural embedding
σ : Sn+1 −→ PC2n ,
where C2n is the spin representation of Spin(2n + 1). Considering now a field of characteristic
two, one obtains similarly an embedding
σ : Sn+1 −→ P2n−1.
It is well-known that the image of σ (and of σ) is defined by quadrics. The spinor variety Sn+1
parametrizes 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 tA = −A (which is A!) and all diagonal coefficients of A should be zero). The maps σ
and σ are given by the 2n Pfaffians of the principal submatrices of A. The restriction of σ to
these subspaces will again be denoted by the same symbol:
σ : An+1 := {alternating (n+ 1)× (n+ 1) matrices} −→ P2n−1.
To show that Zn = σ(Sn+1), we will define in Section 2.4 an explicit map
α : Sn −→ An+1, such that π(Sn) = σ(α(Sn))
for all symmetric matrices Sn ∈ Sn with coefficients in a(ny) algebraically closed field of charac-
teristic 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 3.1.
The map α extends to a map
α : LG(n, 2n) −→ Sn+1.
To complete the picture, we discuss in Section 5 a classical map, over fields of characteristic two,
β : Sn+1 −→ LG(n, 2n), βα = FLG(n,2n), αβ = FSn+1 ,
where F is the Frobenius map, which is induced by the map (. . . : xi : . . .) 7→
(
. . . : x2
i : . . .
)
on
the projective spaces. This points to the ‘exceptional’ isogeny of linear algebraic groups between
Spin(2g + 1) and Sp(2g) in characteristic two as the ‘reason’ for these results.
In fact, after having completed a first draft of this paper, we became aware of the paper [17],
where R. Gow uses this isogeny to provide the ingredients for a more intrinsic proof of the fact
that Zn = σ(Sn+1) in characteristic two, see Remark 5.2. We also noticed the recent paper [28]
which involves the geometry studied in this paper.
Since the Cayley hyperdeterminant, LG(3, 6) and S6 also appear in the context of Freudenthal
triple systems and four-dimensional Maxwell–Einstein supergravity on four space-time dimen-
sions (as well as in [20, Table 3]), we add a brief discussion on some characteristic two aspects
of that topic.
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 3
2 The maps
2.1 The fields
Even if our motivation comes from algebra and geometry over the Galois field F2 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 F2 = 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.
2.2 Principal minors of symmetric matrices
We recall the basics of the principal minors of a symmetric matrix. Let
Sn :=
x11 x12 x13 . . . x1n
x12 x22 x23 . . . x2n
x13 x23 x33 . . . x3n
...
...
...
...
...
x1n x2n x3n . . . xnn
be a symmetric n × n matrix. For a subset I := {i1, . . . , ik} of {1, . . . , n} with 1 ≤ i1 <
· · · < ik ≤ n, the principal minor defined by I is the determinant of the submatrix of Sn with
coefficients (Sn)ij with i, j ∈ I. This principal minor will be denoted by Sn,I and if I is the
empty set we put Sn,∅ = 1. For example,
Sn,∅ = 1, Sn,{i} = xii, Sn,{i,j} = xiixjj − x2
ij ,
Sn,{i,j,k} = xiixjjxkk − xiix2
jk − xjjx2
ik − xkkx2
ij + 2xijxikxjk.
The principal minor map π : Sn → P2n−1 is defined by the
2n =
(
n
0
)
+
(
n
1
)
+ · · ·+
(
n
m
)
+ · · ·+
(
n
n
)
principal minors of the m×m principal submatrices with 0 ≤ m ≤ n.
Example 2.1 (the case n = 3). Let zabc, with a, b, c ∈ {0, 1} be the coordinates on P7. The
principal minor map
π : S3 −→ P7, S3 7−→ (z000 : z001 : . . . : z111) = (S3,∅ : S3,{1} : . . . : S3,{1,2,3}),
in general, zabc = S3,I where 1 ∈ I iff c = 1, 2 ∈ I iff b = 1 and 3 ∈ I iff a = 1. So
z100 = S3,{3} = x33 and z011 = S3,{1,2} = x11x22 − x2
12. The equation of the (Zariski closure of
the) image of π is H = 0 where H is the hyperdeterminant [19, 33])
H := z2
000z
2
111 + z2
001z
2
110 + z2
010z
2
101 + z2
100z
2
011
− 2(z000z001z110z111 + z010z011z100z101 + z000z010z101z111
+ z001z011z100z110 + z000z011z100z111 + z001z010z101z110)
+ 4(z000z011z101z110 + z001z010z100z111).
In case we work over a field of characteristic two, H is the square of a degree two polynomial
H ≡ z2
000z
2
111 + z2
001z
2
110 + z2
010z
2
101 + z2
100z
2
011
≡ (z000z111 + z001z110 + z010z101 + z100z011)2 mod 2,
4 B. van Geemen and A. Marrani
since now (a + b)2 = a2 + 2ab + b2 = a2 + b2. The (closure of the) image Z3 of the map
π : S3 → PF3
2 is defined by this degree two polynomial, since
1 ·
(
x11x22x33 + x11x
2
23 + x22x
2
13 + x33x
2
12
)
+ x11
(
x22x33 + x2
23
)
+ x22
(
x11x33 + x2
13
)
+ x33
(
x11x22 + x2
12
)
= 0.
2.3 Pfaffians of alternating matrices
Let AN = (yij) be the alternating N ×N matrix where the coefficients yij are the variables in
the polynomial ring R := Z[. . . , yij , . . .]1≤i<j≤N (so if j > i then yji = −yij and the diagonal
coefficients of A are zero):
A = AN :=
0 y12 y13 y14 . . . y1N
−y12 0 y23 y24 . . . y2N
−y13 −y23 0 y34 . . . y3N
−y14 −y24 −y34 0 . . . y4N
...
...
...
...
...
...
−y1N −y2N −y3N −y4N . . . 0
.
Then A corresponds to a 2-form
σA :=
∑
1≤i<j≤N
yijei ∧ ej ,
where the ei are the standard basis of RN . In case N is even, one defines a homogeneous
polynomial Pf(A) ∈ Z[. . . , yij , . . .]1≤i<j≤N of degree N/2 by considering the N/2-th exterior
power of σA:
σ
∧N/2
A := σA ∧ σA ∧ · · · ∧ σA︸ ︷︷ ︸
N/2
= (N/2)! Pf(A)e1 ∧ · · · ∧ eN ,
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[. . . , yij , . . .] → K[. . . , yij , . . .]. The
image PfK of the polynomial Pf(A) under this homomorphism defines the Pfaffian of an N ×N
alternating matrix coefficients in K as follows. Let B = (bij) be such an alternating matrix,
then Pf(B) := PfK(. . . , bij , . . .), so we evaluate PfK in yij := bij .
It is not hard to verify the following formula for the Pfaffian of an alternating N×N matrix A
with coefficients yij :
Pf(A) :=
N∑
j=2
(−1)jy1j Pf
(
A1̂ĵ
)
,
where A1̂ĵ 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 yjj = 0):
Pf(A) :=
N∑
j=1
yjn Pf
(
Aĵn̂
)
, char(K) = 2.
For any subset Ĩ ⊂ {1, . . . , N} with an even number of elements we consider the ‘principal’
submatrix of A with coefficients (AN )ab and a, b ∈ Ĩ. These matrices are again alternating and
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 5
thus we can consider their Pfaffians, which we denote by AN,Ĩ and we put AN,∅ := 1. For
example,
AN,∅ = 1, AN,{i,j} = yij , AN,{i,j,k,l} = yijykl − yikyjl + yilyjl.
The Pfaffian map σ : AN → P2N−1−1 is defined by the
2N−1 =
(
N
0
)
+
(
N
2
)
+
(
N
4
)
+ · · ·
Pfaffians of the m×m principal submatrices, with m even and 0 ≤ m ≤ N .
Since ei ∧ ej and ek ∧ el commute in the exterior algebra ∧∗kN , one easily verifies that, with
A = AN ,
exp(σA) := 1 + σA +
1
2!
σA ∧ σA + · · ·+ 1
(N/2)!
σ
∧N/2
A =
∑
Ĩ
Pf(AĨ)eĨ ,
where the sum is over the ordered subsets Ĩ = {i1, . . . , i2k} ⊂ {1, . . . , N} with an even number
of elements and eĨ = ei1 ∧· · ·∧ei2k , since the k! in the definition of Pf(AĨ) cancels with the 1
k! in
the exponential function. Using commutativity as well as (ei ∧ ej)∧2 = (ei ∧ ej) ∧ (ei ∧ ej) = 0,
we also have
exp(σA) = exp
(∑
i<j
yijei ∧ ej
)
=
∏
i<j
exp(yijei ∧ ej) =
∏
i<j
(1 + yijei ∧ ej),
and thus∏
i<j
(1 + yijei ∧ ej) =
∑
Ĩ
Pf(AĨ)eĨ ,
a formula which works over any field, also of finite characteristic. The Pfaffian map now appears
as a natural map from AN into P ∧even KN .
Example 2.2 (the case N = 4). We define the Pfaffian map
σ : A4 −→ P7, A4 7−→ (z000 : z001 : . . . : z111) = (A4,∅ : A4,{1,4} : . . . : A4,{1,2,3,4}),
by zabc = A4,Ĩ and Ĩ is obtained from I with zabc = S3,I in Example 2.1 by Ĩ = I if ]I, the
cardinality of I, is even and else Ĩ = I ∪ {4}. One easily verifies that
A4,∅A4,{1,2,3,4} −A4,{1,2}A4,{3,4} +A4,{1,3}A4,{2,4} −A4,{1,4}A4,{2,3} = 0,
hence the (closure of the image) of σ is the quadric defined by z000z111 − z001z110 + z010z101 −
z100z011.
In particular, the image of σ : A4 → P7 is defined by the degree two polynomial z000z111 +
z001z110 + z010z101 + z100z011 and thus, comparing with Example 2.1, 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).
6 B. van Geemen and A. Marrani
2.4 A map from symmetric to antisymmetric matrices
In Example 2.2 we observed that, over a field with characteristic two, the maps π and σ, with
domains S3 and A4 respectively, have images that are defined by the same quadratic polynomial.
Now we define a map α : Sn → An+1 which will be shown to have the property: π(Sn) = σ(α(Sn))
for any n.
With the notation from Sections 2.2 and 2.3, we define a (non-linear) map
α : Sn −→ An+1, Sn 7−→ S̃n := α(Sn),
(
S̃n
)
ij
=
0 if i = j,
xiixjj + x2
ij if i 6= j, i, j 6= n+ 1,
xii if j = n+ 1,
xjj if i = n+ 1.
Notice that we assume the field to have characteristic two, so S̃n is alternating (in fact,
(
S̃n
)
ii
= 0
for all i and
(
S̃n
)
ij
= −
(
S̃n
)
ji
=
(
S̃n
)
ji
). For example,
α : S3 =
x11 x12 x13
x12 x22 x33
x13 x23 x33
7−→ S̃3 =
0 x11x22 + x2
12 x11x33 + x2
13 x11
x11x22 + x2
12 0 x22x33 + x2
23 x22
x11x33 + x2
13 x22x33 + x2
23 0 x33
x11 x22 x33 0
.
Finally we define how the coordinate functions of π and σ correspond: for any subset I ⊂
{1, . . . , n} we define a subset Ĩ ⊂ {1, . . . , n+ 1} with an even number of elements as follows
Ĩ =
{
I if ]I is even,
I ∪ {n+ 1} if ]I is odd.
We will prove the following theorem in Section 3:
Theorem 2.3. Let π : Sn → P2n−1 be the principal minor map with coordinate functions Sn,I
as in Section 2.2 and let σ : An+1 → P2n−1 be the Pfaffian map with coordinate functions An+1,Ĩ
as in Section 2.3 and where I and Ĩ correspond as above. Let α : Sn → An+1 be defined as in
Section 2.4.
Then we have, over any field of characteristic two
π = σ ◦ α.
In fact, Sn,I = S̃n,Ĩ for all Sn ∈ Sn and all subsets I of {1, . . . , n}.
Examples 2.4. We give some examples of the identity Sn,I = S̃n,Ĩ . Obviously Sn,∅ = 1 = S̃n,∅.
In case I = {i} one has Ĩ = {i, n + 1} and indeed Sn,{i} = xii = S̃n,{1,n+1}. In case I = {i, j}
one has Ĩ = I and we do have the identity
Sn,{i,j} = det
(
xii xij
xij xjj
)
= xiixjj + x2
ij = Pf
(
0 xiixjj + x2
ij
xiixjj + x2
ij 0
)
= S̃n,{i,j}.
Finally if I = {i, j, k} then Ĩ = {i, j, k, n + 1} and we do have Sn,{i,j,k} = Pf
(
S̃n,{i,j,k,n+1}
)
because of the identity
det
xii xij xik
xij xjj xkk
xik xjk xkk
= Pf
0 xiixjj + x2
ij xiixkk + x2
ik xii
xiixjj + x2
ij 0 xjjxkk + x2
jk xjj
xiixkk + x2
ik xjjxkk + x2
jk 0 xkk
xii xjj xkk 0
,
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 7
which holds since
xiixjjxkk + xiix
2
jk + xjjx
2
ik + xkkx
2
ij
=
(
xiixjj + x2
ij
)
xkk +
(
xiixkk + x2
ik
)
xjj +
(
xjjxkk + x2
jk
)
xii.
Notice that these examples show that for n = 3 we have S3,I = S̃3,Ĩ for all subsets I of {1, 2, 3}.
Thus we verified Theorem 2.3 for n = 3 and this will be the starting point for an induction
argument.
3 The proof of Theorem 2.3
3.1 The determinant of a symmetric matrix
In order to prove Theorem 2.3, we start with some observations on the determinant of a sym-
metric matrix, in particular in the case the field has characteristic two.
The determinant of an n× n matrix A = (aij) is
det(A) =
∑
σ∈Σn
sgn(σ)a1σ(1) · · · anσ(n),
where Σn is the symmetric group on {1, . . . , n}. As det(A) = det(tA), under the substitution
aij := aji the monomials of the determinant are either fixed or permuted in pairs. A fixed term
may contain any aii’s and if aij occurs, so does aji. 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 aij , with
i 6= j, occurs in a fixed term, then since aij = aji, the term contains a2
ij . Up to a simultaneous
permutation of the rows and columns (to preserve the symmetry) any term in the determinant
of the symmetric matrix Sn is thus of the form
x11 · · ·xkkx2
k+1,k+2 · · ·x2
n−1,n, k = 0, 1, . . . , n.
Proposition 3.1. Let K be a field of characteristic two and let Sn = (xij) be a symmetric n×n
matrix. Then we have the following relation between principal minors of Sn:
(1) in case n is even,
det(Sn) =
(
x11xnn + x2
1n
)
det(Sn,1̂,n̂) + · · ·+
(
xn−1,n−1xnn + x2
n−1,n
)
det
(
S
n,n̂−1,n̂
)
,
(2) in case n is odd,
det(Sn) =
(
x11xnn + x2
1n
)
det(Sn,1̂,n̂) + · · ·+
(
xn−1,n−1xnn + x2
n−1,n
)
det
(
S
n,n̂−1,n̂
)
+ xnn det(Sn−1),
where det(Sn,̂i,ĵ) is the principal minor Sn,I with I the subset of {1, . . . , n} with only i, j omitted
and Sn−1 = Sn,n̂ is the submatrix of Sn where the last row and column are omitted.
Proof. The right hand sides of the two formulas in Proposition 3.1 are invariant under simul-
taneous permutations of rows and columns which fix the last row and column. Therefore the
formulas follow if the following monomials have equal coefficients on both sides of the identity
tk :=
(
x11 · · ·xkkx2
k+1,k+2 · · ·
)
· x2
n−1,n, t′k :=
(
x11 · · ·xkkx2
k+1,k+2 · · ·x2
n−2,n−1
)
· xnn.
8 B. van Geemen and A. Marrani
Notice that the tk 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 (xiixnn + x2
in) det(Sn,̂i,n̂) and also in xnn det(Sn−1) is
a tk or a t′k up to simultaneous permutation of rows and columns. So we only need to verify
that each term of type tk occurs an odd number of times in the summands on the right hand
sides of Proposition 3.1.
The terms tk all have the variable xn−1,n. In the matrices Sn,̂i,n̂ (i = 1, . . . , n− 1) and Sn−1
appearing in the two formulas in Proposition 3.1 we omit the n-th row and column, so they don’t
have the variable xn−1,n. Only xn−1,n−1xnn +x2
n−1n has this variable. Each tk, 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 tk actually occurs in x2
n−1n · det(S
n,n̂−1,n̂
), provided k has the same parity as n.
Now consider the terms t′k. We notice first of all that t′n−1 = x11 · · ·xnn occurs in all terms
on the right hand side of each of the two formulas in Proposition 3.1 and since the two right
hand sides each have an odd number of terms, it survives.
Next we consider t′n−3 = x11 · · ·xn−3,n−3x
2
n−2,n−1xn,n. Considering x2
n−2,n−1, it obviously
does not occur in the two terms(
xn−2,n−2xnn + x2
n−2,n
)
det
(
S
n,n̂−2,n̂
)
,
(
xn−1,n−1xnn + x2
n−1,n
)
det
(
S
n,n̂−1,n̂
)
.
However, t′n−3 does appear in all other summands of each of the two right hand sides in Propo-
sition 3.1. 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
(
xn−i,n−ixnn +
x2
n−i,n
)
det
(
S
n,n̂−i,n̂
)
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 3.1. �
Proof of Theorem 2.3. We need to show that Sn,I = S̃n,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 Sn,I = S̃n,Ĩ holds for all I ⊂ {1, . . . , n} and we must prove that Sn+1,J = S̃n+1,J̃
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 Sn+1,J = S̃n+1,J̃ 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, J̃ = {1, . . . , n+ 1, n+ 2} and we must show that Sn+1,J = S̃n+1,J̃ , that
is det(Sn+1) = Pf
(
S̃n+1
)
. It is more convenient to change the integer n to n − 1 and then we
must show det(Sn) = Pf
(
S̃n
)
for n odd. Using the formula for computing the Pfaffian given in
Section 2.3 (with N = n+ 1) we have
Pf
(
S̃n
)
=
n+1∑
k=1
(
S̃n
)
k,n
Pf
(
S̃n,k̂,n̂
)
=
(
n−1∑
k=1
(xkkxnn + x2
kn) Pf
(
S̃n,k̂,n̂
))
+ xnn Pf
(
S̃
n,n̂,n̂+1
)
.
The principal submatrix S̃n,n̂ of S̃n obtained by deleting the n-th row and column, is an alterna-
ting n × n matrix where the coefficients xin no longer appear and which is exactly S̃n−1,
so S̃n,n̂ = S̃n−1. For all k ∈ {1, . . . , n − 1} the Pfaffian of the (n − 1) × (n − 1) alterna-
ting matrix S̃n−1,k̂ obtained by deleting the k-th row and column of S̃n−1 is S̃n−1,Ĩ where
Ĩ =
{
1, . . . , k̂, . . . , n
}
. By induction we know that this Pfaffian is det(Sn−1,I) where I ={
1, . . . , k̂, . . . , n−1
}
in case k < n, which is also det
(
Sn−1,k̂
)
. In case k = n, we have
(
S̃n
)
n,n
= 0
and we already omitted this term. Finally if k = n + 1 we have S̃
n,n̂,n̂+1
= S̃n−1,Ĩ where
Ĩ = {1, . . . , n− 1} and thus, by induction, Pf
(
S̃
n,n̂,n̂+1
)
= det(Sn−1). Thus we can rewrite the
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 9
Pfaffian of S̃n in terms of principal minors of Sn−1:
Pf
(
S̃n
)
=
(
n−1∑
k=1
(
xkkxnn + x2
kn
)
det
(
Sn−1,k̂
))
+ xnn det(Sn−1),
and the equality det(Sn) = Pf
(
S̃n
)
for n odd follows from Proposition 3.1(2).
In case n+ 1 is even, J = {1, . . . , n+ 1} = J̃ and we must show that Sn+1,J = S̃n+1,J , that is
det(Sn+1) = Pf
(
S̃
n+1,n̂+2
)
. Again we prefer to change the integer n to n− 1, so we must show
that for n even we have det(Sn) = Pf
(
S̃
n,n̂+1
)
. We have the following expansion of the Pfaffian
of the alternating n× n matrix S̃
n,n̂+1
:
Pf
(
S̃
n,n̂+1
)
=
n−1∑
k=1
(
S̃
n,n̂+1
)
k,n
Pf
(
S̃
n,k̂,n̂,n̂+1
)
=
n−1∑
k=1
(
xkkxnn + x2
kn
)
Pf
(
S̃
n,k̂,n̂,n̂+1
)
.
Notice that S̃
n,k̂,n̂,n̂+1
= S̃n−1,k̂,n̂ and by induction we may assume that
Pf
(
S̃n−1,k̂,n̂
)
= det
(
Sn−1,k̂
)
,
since if n is even, then I :=
{
1, . . . , k̂, . . . , n − 1
}
= Ĩ. Finally we notice that Sn−1,k̂ = Sn,k̂,n̂.
Thus the equality det(Sn) = Pf
(
S̃
n,n̂+1
)
for n even follows from Proposition 3.1(1). �
4 From matrices to Grassmannians
4.1 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.
4.2 The Lagrangian Grassmannian
Let V be a vector space over a field K and let
e : V × V −→ K,
be a symplectic form, that is, an alternating, non-degenerate, bilinear form (so for any x ∈ V ,
e(x, x) = 0 and if x 6= 0, there is a y ∈ V with e(x, y) 6= 0). Then V has a symplectic basis
f1, . . . , f2n, that is, e(fi, fj+n) = −e(fj+n, fi) = δij (Kronecker’s delta) for 1 ≤ i, j ≤ n and all
other e(fi, fj) are zero. So if I denotes the n× n identity matrix, then
e
2n∑
i=1
xifi,
2n∑
j=1
yjfj
=
n∑
i=1
xiyi+n − xi+nyi = (x1 . . . x2n)
(
0 I
−I 0
) y1
...
y2n
.
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 dimW = n, the maximal possible. Choosing a basis w1, . . . , wn
10 B. van Geemen and A. Marrani
of W , let MW be the 2n×n matrix whose columns are the wi. Then W = im
(
MW : Kn → K2n
)
and W is isotropic iff
tMW
(
0 I
−I 0
)
MW = 0 ⇐⇒ tAB − tBA = 0, MW =
(
A
B
)
.
In particular, the subspace W0 := 〈f1, . . . , fn〉 is Lagrangian and MW0 has blocks A = I and
B = 0. More generally, given a symmetric n × n matrix X, the subspace WX spanned by the
columns of the matrix M with blocks A = I and B = X is Lagrangian
Sn ↪→ LG(n, 2n), X 7−→WX := im
(
I
X
)
.
The Lagrangian subspaces of K2n 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-dimen-
sional subspaces of K2n.
4.3 The Plücker map
The Plücker map gives an embedding of
Gr(n, 2n) −→ P ∧n K2n, W 7−→ ∧nW =
∑
I
pI(W )fI ,
where I = {i1, . . . , in} is an ordered subset of {1, . . . , 2n} and fI := fi1∧· · ·∧fin where the fi are
the standard basis of K2n. If W is the span of the columns of an 2n×n matrix MW , then pI(W )
is the determinant of the n× n submatrix of MW given by the rows i1, . . . , in of MW .
To understand the restriction of the Plücker map to the submanifold LG(n, 2n) of Gr(n, 2n),
we recall some general results on the exterior algebra of a symplectic vector space over a field K
of characteristic zero (see [17] and the references given there, or [34, Section 11.6.7], but note the
misprints). Let e be the standard symplectic form on V := K2n, then one defines contraction
maps
∂ : ∧k V −→ ∧k−2V,
∂(v1 ∧ · · · ∧ vk) :=
∑
i<j
e(vi, vj)(−1)i+j−1v1 ∧ · · · ∧ v̂i ∧ · · · ∧ v̂j ∧ · · · ∧ vk.
Let the fi be a symplectic basis of V as before, then we define
ε : ∧k V −→ ∧k+2V, θ 7−→ Γ ∧ θ with Γ :=
n∑
i=1
fi ∧ fi+n
(
∈ ∧2 V
)
.
We extend ∂ and ε to the exterior algebra ∧∗V of V by linearity. Finally we define a linear map
H : ∧∗ V :=
2n⊕
k=0
∧kV −→ ∧∗V, H(θ) = (n− k)θ if θ ∈ ∧kV.
These linear maps define a representation of the Lie algebra sl(2) on ∧∗V :
H = [∂, ε], [H, ∂] = 2∂, [H, ε] = −2ε.
We denote the subspace of highest weight vectors, of weight n−k ≥ 0, for this sl(2)-representation
by (
∧kV
)
0
:=
{
θ ∈ ∧kV : ∂θ = 0
}
, k = 0, 1, . . . , n.
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 11
As a consequence, there is a decomposition ([34, Section 11.6.7, Theorem 3], basically the
Lefschetz decomposition from [18, p. 122]),
∧kV =
⊕
2i≥k−n
Γi ∧
(
∧k−2iV
)
0
,
which is the decomposition of ∧kV into irreducible Sp(2n) subrepresentations. In the case k = n,
the vector space ∧nV is the weight space for sl(2) with weight 0, and thus
∧nV =
(
∧nV
)
0
⊕ V ′n, V ′n = im
(
ε : ∧n−2 V ↪→ ∧nV
)
= im
(
∂ : ∧n+2 V ↪→ ∧nV
)
,
and
(
∧nV
)
0
is a trivial sl(2)-representation, moreover, ε2 : ∧n−2 V → ∧n+2V , ∂2 : ∧n+2 V →
∧n−2V are isomorphisms.
Let W be a Lagrangian subspace of V . Then one can choose a symplectic basis fi for V such
that f1, . . . , fn are a basis of W and one easily finds that now Γ ∧
(
∧nW
)
= 0 ∈ ∧n+2V . Since
the decomposition of ∧nV does not depend on the choice of a symplectic basis we find that
LG(n, 2n) = Gr(n, 2n) ∩P
(
∧nV
)
0
(
⊂P ∧n V
)
,
where we view Gr(n, 2n) as a subvariety of P
(
∧nV
)
.
For example, if n = 3 then LG(3, 6) maps to P13 since the dimension of
(
∧3V
)
0
is then
20− 6 = 14, this case is discussed in [22] and Section 6.7.
4.4 The principal minor map
The principal minor map extends to a map, again denoted by π,
π : LG(n, 2n) −→ P2n−1, W 7−→ (. . . : pJ(W ) : . . .),
where J runs over the 2n 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 MW and MW has blocks I and
X ∈ Sn, then these pJ(W ) are easily seen to be the principal minors of X. Thus π is a projection
of LG(n, 2n) ⊂ P
(
∧nK2n
)
0
into P2n−1 and it is not hard to verify that π is a regular map (base
point free) on LG(n, 2n). The closure Zn of π(Sn) is thus the projective variety π(LG(n, 2n)).
We now show that the morphism π : LG(n, 2n)→ Zn has degree 2n−1, if the characteristic of
the field K is not two. (In the lemma below, LG(n, 2n)/Gn is not isomorphic to Zn for n > 3
since there are invariant monomials in the xij on Sn ⊂ LG(n, 2n) which are not contained in
the ring of principal minors.)
Lemma 4.1. The principal minor map π : LG(n, 2n) → Zn
(
⊂P2n−1
)
has degree 2n−1 over
a field of characteristic different from two. This map factors over a quotient of LG(n, 2n) by
a group Gn ∼= (Z/2Z)n−1.
Proof. Any diagonal matrix D = diag(t1, . . . , tn, t
−1
1 , . . . , t−1
n ) with ti 6= 0 fixes the symplectic
form e and thus maps LG(n, 2n) into itself by W 7→ DW , equivalently, MW 7→ DMW . Let
D1 := diag(t1, . . . , tn), and notice that DMW and DMWD
−1
1 map Kn to the same subspace DW
in K2n. For MW with blocks I, X, the matrix DMWD
−1
1 has blocks I, D−1
1 XD−1
1 , so we see
that D maps the image of Sn in LG(n, 2n) into itself and acts as D : X 7→ D−1
1 XD−1
1 . In
case all ti ∈ {1,−1}, we have D−1
1 = D1 and we write more suggestively D : X 7→ D1XD
−1
1 ,
the conjugation by D1. Any principal submatrix of X is then also conjugated by a submatrix
of D1, and hence the principal minors of X and those of D1XD
−1
1 are the same. So the fiber
of π over π(X) contains all the D1XD
−1
1 where D1 has coefficients ±1. Obviously D1 = −I
acts trivially and thus we have an action of the group Gn := (Z/2Z)n−1 on LG(n, 2n) and π
12 B. van Geemen and A. Marrani
factors over LG(n, 2n)/Gn. The ij-coefficient of D1XD
−1
1 is xijtitj . Since the xii, xiixjj−x2
ij are
principal minors of X, we can recover the xij from π(WX), except for the signs of the xij with
i 6= j. However, the principal minors Sn,{i,j,k} (see Section 2.2) show that once, for a fixed i, all
the xil are non-zero and the signs of all these xil are fixed, then the signs of all xjk are fixed.
Therefore the fiber over π(X), for general X ∈ Sn, consists of exactly 2n−1 elements that are an
orbit of Gn. This implies that π has degree 2n−1 and that π factors over LG(n, 2n)/Gn. �
4.5 The spinor varieties
A quadratic form on a vector space V over a field K is a map
q : V −→ K, such that q(ax) = a2q(x), q(x+ y) = q(x) + q(y) + e(x, y),
where a ∈ K and e is a bilinear form and x, y ∈ V . We consider the quadratic form q on
V = K2n defined by
q
(
2n∑
i=1
xifi
)
:=
n∑
i=1
xixi+n, 2q(x) = (x1 . . . x2n)
(
0 I
I 0
) x1
...
x2n
.
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 dimW = n, the maximum possible.
Choosing a basis w1, . . . , wn of W , let MW be the 2n × n matrix whose columns are the wi.
Then W = im
(
MW : Kn → K2n
)
. The subspace W is maximally isotropic for q iff
q(wi) = 0, q(wi + wj) = 0, 1 ≤ i, j ≤ n,
in fact, if q(wi) = 0 and also 0 = q(wi + wj) = e(wi, wj) for all i, j, then from
q
(
n∑
i=1
aiwi
)
= q
(
n−1∑
i=1
aiwi
)
+ a2
nq(wn) +
n−1∑
i=1
aiane(wi, wn)
=
n∑
i=1
a2
i q(wi) +
∑
i<j
aiaje(wi, wj)
we see that W is maximally isotropic. In case char(K) 6= 2 this can also be checked using the
symmetric matrix of e:
tMW
(
0 I
I 0
)
MW = 0 ⇐⇒ tAB + tBA = 0, MW =
(
A
B
)
,
and notice that q(wi) =
(
tAB + tBA
)
ii
and q(wi + wj) =
(
tAB + tBA
)
ij
.
The subspace W0 := 〈f1, . . . , fn〉 is thus maximally isotropic for q. More generally, given an
antisymmetric n×n matrix Y , the subspace WY spanned by the columns of the matrix M with
blocks A = I and B = Y is Lagrangian, so
An ↪→ S+
n , Y 7−→ im
(
I
Y
)
,
where S+
n denotes the spinor variety containing W0. This holds over any field, since q(wi) = yii
and q(wi + wj) = yii + yjj + yij + yji and thus WY is maximally isotropic for q iff the diagonal
coefficients of Y are zero and yij+yji = 0 iff Y is alternating. Recall that there are two n(n−1)/2-
dimensional families of maximally isotropic subspaces of q. They are parametrized by the spinor
varieties S+
n and S−n , which are isomorphic. For spinor varieties see [8], [34, Section 11.7] and
the references given in [35, Section 6.0].
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 13
4.6 The image of the Pfaffian map
Over the complex numbers, the Pfaffian map on An from Section 2.3 extends to an embedding
of the spinor variety
σ : S+
n −→ P2n−1−1.
In the introduction we used a map σ on the spinor variety associated to Spin(2n − 1), but we
will see that these spinor varieties are isomorphic in Section 5.2.
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
[34, Section 11.7.2], σ(Sn) is also the G-orbit 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 e1, . . . , en is a basis of W , the
element exp(yijei∧ej) =
∏
(1+yijei∧ej) introduced in Section 2.3 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 [37]. This implies that the
image of σ is an intersection of quadrics. The number of quadrics can also be determined, it is
dim I2 :=
(
2n−1 + 1
2
)
− 1
2
(
2n
n
)
, I2 :=
{
Q ∈ k[. . . , zI , . . .] : Q(σ(W )) = 0 ∀W ∈ S+
n
}
,
where K[. . . , zI , . . .] is the homogeneous coordinate ring of P2n−1−1, in fact, [37] shows that
dim I2 does not depend on the characteristic of the field and over the complex numbers one
can use for example (the proof of) [16, 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 [34, Sec-
tion 11.7.2] for the quadratic relations between Pfaffians, [39] for explicit methods to find the
quadratic equations of σ
(
S+
n
)
and [35, Section 6] for a study of the case n = 5.
Proposition 4.2. Let π : Sn → P2n−1 be the principal minor map over an algebraically closed
field of characteristic two. Then the closure Zn of the image of π is σ
(
S+
n+1
)
and in particular Zn
is an intersection of quadrics.
Proof. Since the symmetric matrices Sn are Zariski dense in LG(n, 2n) and the alternating
matrices are Zariski dense in S+
n+1, we find, using Theorem 2.3, that
Zn = π(LG(n, 2n)) = σ
(
S+
n+1
)
⊂ P2n−1.
In Section 4.6 we recalled that σ
(
S+
n+1
)
is defined by quadrics, hence also Zn is defined by
quadrics. �
5 The map β
5.1 From antisymmetric to symmetric matrices
We work over a field of characteristic two. In Section 2.4 we defined α : Sn → An+1 in such
a way that the principal minors of Sn were the Pfaffians of α(Sn), this condition determined
the map α. Now we consider a map β : An+1 → Sn, which is defined in terms of a well-known
map from S+
n+1
∼= Sn+1 → LG(n, 2n), which we will also denote by β. The maps α and β are
14 B. van Geemen and A. Marrani
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 An+1 = (yij) ∈ An+1 be an alternating (n+ 1)× (n+ 1) matrix (so yii = 0 and yij = yji)
and define
β : An+1 −→ Sn, An+1 7−→ An+1 := β(An+1),
(
An+1
)
ij
:= yij + yi,n+1yj,n+1.
For example,
A4 =
0 y12 y13 y14
y12 0 y23 y24
y13 y23 0 y34
y14 y24 y34 0
7−→ A4 =
y2
14 y12 + y14y24 y13 + y14y34
y12 + y14y24 y2
24 y23 + y24y34
y13 + y14y34 y23 + y24y34 y2
34
.
It is not hard to verify that
β(α(Sn))ij = (Sn)2
ij , α(β(An+1))kl = (An+1)2
kl,
for all i, j = 1, . . . , n and all k, l = 1, . . . , n+ 1. Thus the maps βα : Sn → Sn and αβ : An+1 →
An+1 are the (coordinate wise) Frobenius maps on the respective vector spaces of matrices
βα = FSn , αβ = FAn+1 .
5.2 From even to odd spinor varieties
We denote the field of characteristic two by K. In Section 4.5 we considered an embedding
An+1 ↪→ S+
n+1, where S+
n+1 parametrizes certain maximally isotropic subspaces for the quadratic
form q(y) =
n+1∑
i=1
yiyn+1+i on K2n+2. We define a hyperplane
H : yn+1 + y2n+2 = 0
(
⊂K2n+2
)
.
The intersection H ∩ (q = 0) can be identified with the quadric in K2n+1 defined by q′,
q′ = q|H : K2n+1 −→ K, q′(z) = z1zn+2 + · · ·+ znz2n+1 + z2
n+1,
simply by mapping z = (z1, . . . , z2n+1) 7→ y = (z1, . . . , z2n+1, zn+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 dimW = 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 an isomorphism
S+
n+1
∼=−→ Sn+1
between the spinor variety of Spin(2n + 2) containing W0 as in Section 4.5 and the spinor
variety Sn+1 of Spin(2n+ 1) that parametrizes the maximally isotropic subspaces for q′.
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 15
5.3 The geometry of β
We explain the geometry behind the map β, using the notation from Section 5.2. Since the
field K is assumed to have characteristic two, the alternating bilinear form e′ defined by q′ is
degenerate
e′(z, w) := q(z + w) + q(z) + q(w),
e′(z, w) = z1wn+2 + · · ·+ znw2n+1 + zn+2w1 + · · ·+ z2n+1wn,
since the variables zn+1, wn+1 do not appear in e′. More intrinsically, define the subspace
ker(e′) :=
{
z ∈ K2n+1 : e′(z, w) = 0 ∀w ∈ K2n+1
}
,
then we see that ker(e′) is one-dimensional (and is spanned by the n + 1-st standard basis
vector of K2n+1). We consider the quotient space K2n+1/ ker(e′) ∼= K2n where we map z =
(z1 . . . , z2n+1) 7→ x = (z1, . . . , ẑn+1, . . . , z2n+1), i.e., we omit the (n+ 1)-st coefficient. Since e′ is
bilinear, it defines a non-degenerate alternating form ē on this quotient space simply by defining
ē(x, y) := e′(z, w), where z, w ∈ K2n+1 map to x, y ∈ K2n respectively.
Let W ′ be a maximally isotropic subspace in q′ = 0. Then for z, w ∈ W ′ ⊂ (q′ = 0) we have
e′(z, w) = 0, and thus W ′, the image of W ′ in K2n, is an isotropic subspace for the symplectic
form ē on K2n. Since ker(e′) ∩ (q′ = 0) = {0}, the projection W ′ also has dimension n and
hence W ′ is a Lagrangian subspace for the symplectic form ē on K2n. Thus we have a map
β : Sn+1 −→ LG(n, 2n), W ′ 7−→W ′.
It is well-known that the orthogonal group O(q) of q acts as the identity on ker(e′) and that the
projection to K2n induces a homomorphism (an isogeny) of algebraic groups O(q) → Sp(2n),
the symplectic group defined by ē ([38, Section 4.11], [31], [9, Section 7.1, Remark 7.1.6]).
Proposition 5.1. The map β : Sn+1 → LG(n, 2n) we just defined induces the map β : An+1 →
Sn from Section 5.1.
Proof. Given Y ∈ An+1, let WY ⊂ K2n+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∩WY is spanned by
the n vectors ci+yi,n+1cn+1, i = 1, . . . , n, where ci is the i-th column of M , in fact the n+1 and
2n+ 2 coefficients of ci+ yi,n+1cn+1 are 0 + yi,n+1 ·1 and yi,n+1 + yi,n+1 ·0 respectively, and their
sum is indeed zero, showing that these vectors do lie in H ∩WY . Next we project these vectors
to K2n, so we omit the (n+ 1)-st coefficients, their span is then H ∩WY . The image vectors are
the columns of the 2n×n matrix with blocks I and Y , which proves that β induces Y 7→ Y . �
Remark 5.2. 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 Bn and Cn respectively) over a field
of characteristic two, cf. [38, Section 4.11], [31], [9, Section 7.1, Remark 7.1.6]. The description
of the isogeny leads directly to the map β : Sn+1 → LG(n, 2n).
Using this isogeny, in [25, p. 197] one finds the definition of the (2n-dimensional) Spin rep-
resentation of Sp(2n)(K), where K is a field of characteristic two. In Section 4.3 we recalled
the decomposition of ∧kV into Sp(2n)-representations, where V is the standard 2n-dimensional
representation of Sp(2n) in case the field K has characteristic zero.
Now we assume that the field K has characteristic two. The symplectic form on V still induces
an Sp(2n)-equivariant contraction map ∂n : ∧nV → ∧n−2V and
(
∧nV
)
0
:= ker(∂n) is an Sp(2n)-
subrepresentation of ∧nV (but if n > 3, then the dimension of
(
∧nV
)
0
in characteristic two is
larger than its dimension in characteristic zero, cf. [17, Theorem 2.2]). Gow [17] showed that
now the image of the contraction map ∂n+2 : ∧n+2 V → ∧nV has codimension 2n in ker(∂n) and
16 B. van Geemen and A. Marrani
that the quotient Sp(2n)-representation ker(∂n)/ im(∂n+2) is the Spin representation of Sp(2n).
The explicit description of the quotient module given in [17, Section 3] shows also that the
composition LG(n, 2n)→ P
((
∧nV
)
0
)
→ P(ker(∂n)/ im(∂n+2)) factors over the principal minor
map π : LG(n, 2n)→ P2n−1.
Since the composition π of the Plücker map with Gow’s quotient map
LG(n, 2n) −→ P
(
∧nV
)
0
−→ P(ker(∂n)/ im(∂n+2)) = P2n−1
is equivariant for the action of Sp(2n) and Sp(2n) acts transitively on LG(n, 2n), this composition
is everywhere defined (so there are no base points). The image of LG(n, 2n) is then the unique
closed orbit of Spin(2n+ 1) in the projectivization of its spin representation, which is the spinor
variety Sn+1. Thus we get a map LG(n, 2n)→ Sn+1 ‘for free’.
So, just from the isogeny and Gow’s results, we recover a main result which we previously
deduced from explicit computations. However, if one would like to know what the restriction
of this map to the subset of LG(n, 2n) parametrized by symmetric matrices is and what the
quadratic relations between principal minors in characteristic two are, then the results of the
first part of this paper are still useful.
6 Freudenthal triple systems
6.1 Outline
The case n = 3, where we considered Cayley’s hyperdeterminant and the Lagrangian Grass-
mannian LG(3, 6), and the case n = 6, where we considered the spinor variety S6 ⊂ P31
associated to Spin(12), appear in the context of groups of type E7 [6] and reduced Freudenthal
triple systems related to cubic Jordan algebras, as well as in the Freudenthal magic square [27].
These subjects are also of relevance in Maxwell–Einstein four-dimensional supergravity (see, e.g.,
[1, 4, 7, 11, 12, 14]), as well as in the so-called black-hole/qubit correspondence (cf. [2, 5, 10, 24]).
It should be noticed that one usually excludes fields of characteristic 2 and 3 in these subjects.
However, e.g., in [3] 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.
6.2 Groups of type E7
A group of type E7 is the subgroup, which we denote by G4, 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. [6, 26]). It turns out that given some additional
conditions, including a compatibility between e and q and char(k) 6= 2, 3, as well as the condition
that −1
2q(f, f, f, f) is a non-zero square for some f ∈ R (such triples (R, q, e) are called reduced,
[6, p. 90]), the vector space R decomposes as
R = k ⊕ J ⊕ J ⊕ k,
where J is a (cubic) Jordan algebra [23], in [26, 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
A :=
a z ȳ
z̄ b x
y x̄ c
, a, b, c ∈ k, x, y, z ∈ C,
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 17
where C is a composition algebra with involution x 7→ x̄. 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 [26, Section 2.1]
here.
Notice that the dimension of J as a k-vector space is 3q + 3 where q := dimC and dimR =
2+2(3q+3) = 6q+8. Over the complex numbers (the split case), the group G4 will be SL(2,C)3,
Sp(6,C), SL(6,C), Spin(12,C) and E7(C) for q = 0, 1, 2, 4, 8, respectively, cf. [20, Table 3]. The
G4-orbit Yq of the highest weight vector in PR = P6q+7 is a complex projective algebraic
variety of dimension 3q+ 3, it is the unique closed orbit of G4 in PR. The tangential variety Xq
of Yq (see also Section 6.6) has dimension 6q + 6 and is defined by the quartic polynomial q
(cf. [11, 20, 27]).
6.3 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− axx̄− byȳ − czz̄ + (xy)z + z̄(ȳx̄).
One defines the regular trace Tr: J → k and a symmetric bilinear form (−,−) : J × J → k by
Tr(A) = a+ b+ c, (A,B) := Tr
(
1
2(AB +BA)
)
.
Finally there is ‘sharp’ operation on J , similar to the adjoint of a matrix,
A] :=
bc− n(x) ȳx̄− cz zx− bȳ
xy − cz̄ ac− n(y) z̄ȳ − ax
x̄z̄ − by yz − ax̄ ab− n(z)
.
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. [26, equation (29)]). With these definitions (cf. [26, Section 2.4]),
the alternating form e : R×R→ k is given by (cf. [26, Section 3.1])
e((a,A,B, b), (c, C,D, d)) = ad− bc+ (A,D)− (B,C),
and the quartic form q : R→ k is
G := −2G′, G′((a,A,B, b)) = −4
(
A], B]
)
+ 4aN(A) + 4bN(B) +
(
(A,B)− ab
)2
.
A Freudenthal triple system is a vector space R with a non-degenerate alternating form and
a composition T : R×R×R→ R such that certain conditions are satisfied cf. [26, Section 3.1].
Under additional conditions one recovers a group of type E7 as the automorphism group of
a Freudenthal triple system.
Example 6.1 (q = 0). Recall that q := dimC and if C = 0 then J is the three-dimensional
algebra of diagonal matrices and dimR = 8. We change the coordinates as follows
x := (a,diag(a11, a22, a33),diag(b11, b22, b33), b) = (z000, z110, z101, z011, z001, z010, z001, z111),
and similarly ((c, C,D, d) = (y000, . . . , y111). The symplectic form is then
e(x, y) = z000y111 − z001y110 − z010y101 + z011y100 − z100y011 + z101y010
+ z110y001 − z111y000
and the quartic polynomial G′ is the hyperdeterminant: G′(x) = H (cf. Example 2.1). Since the
hyperdeterminant becomes a square modulo two, as we observed in Example 2.1, we consider
some other cases of the construction above.
18 B. van Geemen and A. Marrani
6.4 Reduction modulo two
More generally, if we assume that N(A), N(B) and (A,B) are polynomials in the coefficients
of A, B with integer coefficients, then the reduction of the quartic form G′ is
G′ ≡ Q2 mod 2, Q((a,A,B, b)) := (A,B)− ab,
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 [3, 13].
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(
(A+ C,B +D)− (a+ c)(b+ d)
)
−
(
(A,B)− ab
)
−
(
(C,D)− cd)
)
= (A,D) + (B,C)− ad− bc,
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 [26, Remark 20]. The case q = 1 presents some extra features
since in that case the quadric and the alternating form are degenerate, see Example 6.2.
Example 6.2 (q = 1). We consider the constructions from Section 6.3 for the case that k =
C = R, with x̄ = x and n(x) = x for x ∈ C. Then q = dimk C = 1 and dim J = 6, dimR = 14
and R =
(
∧3k6
)
0
, which is an irreducible Sp(6,R)-representation, cf. Section 4.3. 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 = (aij), . . ., D = (dij) where A, . . . ,D are 3× 3 symmetric
matrices. Then
e(((a,A,B, b), (c, C,D, d)) = a11d11 − b11c11 + a22d22 − b22c22 + a33d33 − b33c33
+ 2
(
a12d12 − b12c12 + a13d13 − b13c13 + a23d23 − b23c23) + ad− bc,
notice the factors 2 which appear. The quartic G′(x) = a2
11b
2
11 + 4a11a22a33a+ · · ·+ a2b2 has 44
terms. Reducing modulo two one finds
G′ ≡ Q2, Q := a11b11 + a22b22 + a33b33 + ab,
and Q = 0 is a singular quadric in P13 (notice that the 6 = 3 + 3 variables aij , bij 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 [26] discusses the next three cases, q = 2, 4, 8,
the case q = 1 is avoided.
As we observed in Remark 5.2, Gow showed that the Sp(6)-representation R has a subrepre-
sentation im(∂5), the image of the contraction map ∂5 : ∧5 k6 → ∧3k6, with 23 = 8-dimensional
quotient R/ im(∂5), the Spin representation of Sp(6). In this case, the subrepresentation coin-
cides with the kernel of the bilinear form defined by Q, written again as Q, that is
im(∂5) = {v ∈ R : Q(v, w) = 0 ∀w ∈ R}.
Thus Q restricts to a nondegenerate quadratic form on the Spin representation of Sp(6). Ac-
cording to [17, Section 4], such an Sp(2n)-invariant non-degenerate quadratic form on the Spin
representation exists for any n.
Lagrangian Grassmannians and Spinor Varieties in Characteristic Two 19
6.5 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 [40]), we
use the expression for the quartic invariant that we found in [15]. Let X := (xij), Y := (yij) be
alternating 8× 8 matrices over the real or complex numbers. We view the xij , yij as coordinates
on a 28+28 = 56-dimensional vector space and we define a symplectic form on this vector space
by requiring that the xij , yij are the coordinates on a symplectic basis. The quartic invariant
of E7 is defined as
J := Pf(X) + Pf(Y )− 1
4 Tr(XYXY ) + 1
16(Tr(XY ))2.
The degree four polynomial J has 1036 terms and coefficients in
{
±1,±1
2 ,−
1
4
}
. To find a re-
duction mod 2 we simply multiply J by 4 and then reduce mod 2 to obtain a quartic J which
has only 28 terms
J = x2
12y
2
12 + x2
13y
2
13 + · · ·+ x2
78y
2
78 =
∑
1≤i<j≤28
x2
ijy
2
ij ,
which is the square of the quadratic polynomial Q =
∑
i<j
xijyij . Notice that the alternating form
defined by Q is indeed the symplectic form defined above.
6.6 Tangential varieties
The representation of the group G4 on PR = P6q+7 has a unique closed orbit Yq of dimension
3q + 3 and one recovers the zero locus of the quartic invariant as the (6q + 6)-dimensional
tangential variety Xq of Yq. This allows one to determine the equation of Xq over a field of
characteristic two. In the case q = 1 we again find the singular quadric from Example 6.2 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 PN , 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 [41]. If locally we have a parametrization
φ : kn −→ Z, φ(x) = (f0(x) : . . . : fN (x))
for certain functions on kn, then the tangent spaces to the image of φ are locally parametrized by
φ̃ : kn × kn −→ PN , (x, y) 7−→
. . . : fk(x) +
n∑
j=1
∂fk
∂xj
(x)yj : . . .
.
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 [32].
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).
One of the referees for this paper offered the following explanation (actually we simplify quite
a bit) for this ‘bad’ behaviour. In the examples we consider the secant variety of Yq, which is the
closure of the union of all lines in PR joining two distinct points in Yq, is all of PR. Moreover,
a general point of PR lies on a unique secant of Yq. Over an open subset U ⊂ PR we thus
have a fibration L → U where the fiber Lr over r ∈ U is the unique secant containing r. On
each secant there are the two points where Lr intersects Yq. These points give a subvariety
20 B. van Geemen and A. Marrani
V ⊂ L and the induced map π : V → U has degree two. The map π ramifies exactly when Lr
is tangent to Yq, that is when r ∈ Xq, the tangential variety of Yq. Notice that locally V is
defined by an equation a(r)s2 + b(r)st+ c(r)t2 with r ∈ U and (s : t) homogeneous coordinates
on the line Lr. The ramification locus of π is then defined by b2(r) − 4a(r)c(r) = 0 which
reduces to b(r)2 = 0 when the field has characteristic two. The examples in the next section
in fact show that the quartics defining the tangential variety Xq are the squares of quadrics
in that case. A worked out example for a related, but simpler, case q = −1 is given in [36,
Example 4.7].
6.7 The various cases
We verified in the cases q = 0, 1, 2, 4 that the zero locus Xq of the quartic invariant, which is the
tangential variety of the closed orbit Yq, is actually a quadric in characteristic two, by explicitly
finding an equation for the tangential variety to Yq in characteristic two.
In case q = 0, we verified that the quadric we found in Example 6.1 is the tangential variety
of the Segre 3-fold Y0, the image of P1 ×P1 ×P1 in P7.
For q = 1, one finds by direct computation that the image Y1 of LG(3, 6) under the Plücker
map spans a P13 ⊂ P19. The tangential variety X1 of LG(3, 6) in this P13 is a singular quadric
of rank 8. The singular locus of the quadric is a P5 ⊂ P13. Notice that this P5 is mapped into
itself under the action of Sp(6) on LG(3, 6) and P13, in fact P5 is P(im ∂5), the subrepresentation
of PR as in Example 6.2.
For q = 2, we found that the tangential variety of the Plücker embedded Y2 = Gr(3, 6) ⊂ P19
is a smooth quadric.
Also for q = 4, we checked that the tangential variety of the 15-dimensional Y4 = S6, Pfaffian
embedded in P31, is a smooth quadric (cf. [29] for such aspects of the geometry of spinor
varieties).
In case q = 8, the variety Y8 of dimension 27 in P55 is known as the Freudenthal variety [30].
Its tangential variety X8, over the complex numbers, is the quartic hypersurface defined by
J = 0 in Section 6.5. We haven’t computed what happens over a field of characteristic two since
the only parametrization of Y8 that we know of is rather cumbersome.
Acknowledgments
BvG would like to thank L. Oeding and W. van der Kallen for helpful correspondence and
discussions. We are indebted to the referees of this paper for comments and suggestions for
improvements.
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22 B. van Geemen and A. Marrani
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1 Introduction
2 The maps
2.1 The fields
2.2 Principal minors of symmetric matrices
2.3 Pfaffians of alternating matrices
2.4 A map from symmetric to antisymmetric matrices
3 The proof of Theorem 2.3
3.1 The determinant of a symmetric matrix
4 From matrices to Grassmannians
4.1 Global aspects
4.2 The Lagrangian Grassmannian
4.3 The Plücker map
4.4 The principal minor map
4.5 The spinor varieties
4.6 The image of the Pfaffian map
5 The map
5.1 From antisymmetric to symmetric matrices
5.2 From even to odd spinor varieties
5.3 The geometry of
6 Freudenthal triple systems
6.1 Outline
6.2 Groups of type E7
6.3 The alternating form and the quartic
6.4 Reduction modulo two
6.5 The cases q=2,4,8
6.6 Tangential varieties
6.7 The various cases
References
|
| id | nasplib_isofts_kiev_ua-123456789-210231 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:50Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | van Geemen, B. Marrani, A. 2025-12-04T13:04:22Z 2019 Lagrangian Grassmannians and Spinor Varieties in Characteristic Two / B. van Geemen, A. Marrani // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14M17; 20G15; 51E25 arXiv: 1903.01228 https://nasplib.isofts.kiev.ua/handle/123456789/210231 https://doi.org/10.3842/SIGMA.2019.064 The vector space of symmetric matrices of size n has a natural map to a projective space of dimension 2ⁿ −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 the spinor variety associated to Spin(2n+1). 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 other cases related to integral Freudenthal triple systems over integral cubic Jordan algebras. BvG would like to thank L. Oeding and W. van der Kallen for helpful correspondence and discussions. We are indebted to the referees of this paper for comments and suggestions for improvements. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Lagrangian Grassmannians and Spinor Varieties in Characteristic Two Article published earlier |
| spellingShingle | Lagrangian Grassmannians and Spinor Varieties in Characteristic Two van Geemen, B. Marrani, A. |
| title | Lagrangian Grassmannians and Spinor Varieties in Characteristic Two |
| title_full | Lagrangian Grassmannians and Spinor Varieties in Characteristic Two |
| title_fullStr | Lagrangian Grassmannians and Spinor Varieties in Characteristic Two |
| title_full_unstemmed | Lagrangian Grassmannians and Spinor Varieties in Characteristic Two |
| title_short | Lagrangian Grassmannians and Spinor Varieties in Characteristic Two |
| title_sort | lagrangian grassmannians and spinor varieties in characteristic two |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210231 |
| work_keys_str_mv | AT vangeemenb lagrangiangrassmanniansandspinorvarietiesincharacteristictwo AT marrania lagrangiangrassmanniansandspinorvarietiesincharacteristictwo |