§§ §§

¢

its dimension is n . Note that k V 0 if k n rk V .

3 ¥ 5 ¥

&

k4 ¤

The exterior algebra of V is 6

n

#

¢ ¢ k

V V

¥¤ & ¦¤

k$ 0

with multiplication de¬ned as for the tensor algebra. Its rank is ‘n 0 n 2n . 3 ¥

k$ k4

If θ is a linear transformation on V , then θ induces in a natural way linear ¢

transformations k θ on k V , and k θ on k V , for all k. If rk V n, then we

" " & & ¥¤

¢ ¢

θ is a scalar. In fact, θ det θ . (This fact is the

n n n

have rk V 1, and so

& ¥¤ & & ¥ ¤

basis of an abstract, matrix-free, de¬nition of the determinant.)

Exercises

1. Let F be a skew ¬eld, V a right F-vector space, and W a left vector space.

Show that it is possible to de¬ne V F W as an abelian group so that

¢ ¢

v1 v2 w v1 w v2 w¦ v w1 w2 v w1 v w2

¥ ¥¤

£ ¤ £ £ £

and ¢ ¢

±w

v± w v

¥

¤ §¨¤

2. In the identi¬cation of F n F m with the space of n m matrices, show

that the rank of a matrix is equal to the minimum r for which the corresponding

tensor can be expressed in the form ‘r 1 vi wi . Show that, in such a minimal

i$

expression, v1 vr are linearly independent, as are w1 wr .

¦ § ¦ ¦ 2§ ¦

§§ §§

3. (a) If K is an extension ¬eld of F, and n a positive integer, prove that

¢ ¢

Mn F K Mn K

F 7¥ ¦¤

¤

¢

where Mn F is the ring of n n matrices over F.

¤

@

9

(b) Prove that . 8 8 8CAB8

7¥

4. De¬ne the symmetric square S2V of a vector space V , the free-est bilinear

square of V in which v w w v. Find a basis for it, and calculate its dimension.

¥

More generally, de¬ne the kth symmetric power SkV , and calculate its dimension; ¢ ¢

and de¬ne the symmetric algebra S V . If dim V n, show that the symmetric ¥¤

¤

algebra on V is isomorphic to the polynomial ring in n variables over the base

¬eld. ¢ ¢

5. Prove that, if θ is a linear map on V , where rk V n, then n θ det θ . ¥¤ ¥

& ¤

150 10. Exterior powers and Clifford algebras

10.2 The geometry of exterior powers

Let V be an F-vector space of rank n, and k a positive integer less than n. There ¢

are a couple of ways of de¬ning a geometry on the set Σk Σk V of subspaces ¥ ¤

¢ ¢

of V of rank k (equivalently, the k 1 -dimensional subspaces of PG n 1¦ F , 3

k4 © ¤ © ¤

which I now describe.

The ¬rst approach produces a point-line geometry. For each pair U1 U2 of ¦

¢ ¢

subspaces of V with U1 U2 , rk U1¤ k 1, rk U2 k 1, a line ¥ ¥¤

D © £

¢

Σk : U1

L U1 U2¤ W W U2

¦ ¥ ¡ D D §

Now two points lie in at most one line. For, if W1 W2 are distinct subspaces of ¦

¢

rank k and W1 W2 L U1 U2 , then U1 W1 W2 and W1 W2 U2 ; so equality

¦ ¡ ¦ G ¦

¤ E F IH

E

must hold in both places. Note that two subspaces are collinear if and only if

their intersection has codimension 1 in each. We call this geometry a subspace

geometry. ¢

In the case k 2, the points of the subspace geometry are the lines of PG n

¥ ©

1¦ F , and its lines are the plane pencils. In particular, for k 2, n 4, it is the ¥ ¥

¤

Klein quadric.

The subspace geometry has the following important property:

Proposition 10.1 If three points are pairwise collinear, then they are contained

in a projective plane. In particular, a point not on a line L is collinear with none,

one or all points of L.

Proof Clearly the second assertion follows from the ¬rst. In order to prove the

¬rst assertion, note that there are two kinds of projective planes in the geometry,

consisting of all points W (i.e., subspaces of rank k) satisfying U1 W U2 , D D

¢ ¢ ¢ ¢

where either rk U1¤ k 1, rk U2 k 2, or rk U1¤ k 2, rk U2 k 1.

¥ ¥¤ ¥ ¥¤

© £ © £

¢

So let W1 W2 W3 be pairwise collinear points. If rk W1 W2 W3 k 1,

¦ ¦ ¥¤

F F ©

then the three points are contained in a plane of the ¬rst type; so suppose not.

¢

Then we have rk W1 W2 W3 k 2; and, by factoring out this intersection, ¥¤

F F ©

we may assume that k 2. In the projective space, W1 W2 W3 are now three

¥ ¦ ¦

pairwise intersecting lines, and so are coplanar. Thus rkG W1 W2 W3 k 1, and ¦ ¦ ¥H £

our three points lie in a plane of the second type.

A point-line geometry satisfying the second conclusion of Proposition 10.1 is

called a gamma space. Gamma spaces are a natural generalisation of polar spaces

10.2. The geometry of exterior powers 151

(in the Buekenhout“Shult sense), and this property has been used in several recent

characterisations (some of which are surveyed by Shult [29]).

The subspace geometries have natural embeddings in projective spaces given

k

by exterior powers, generalising the Klein quadric. Let X V ; we consider P¥

&

¢ n

the projective space PG N 1¦ F based on X , where N k4 . This projective 3¥

© ¤

space contains some distinguished points, those spanned by the vectors of the

form v1 vk , for v1 vk V . We call these pure products.

' § ' ¦ 2§ ¦ ¡

§§ §§

Theorem 10.2 (a) v1 vk 0 if and only if v1 vk are linearly depen-

' ' 2§ ¥ ¦ § ¦

§§ §§

dent.

¢

(b) The set of points of PG N 1¦ F spanned by non-zero pure products, together

© ¤

with the lines meeting this set in more than two points, is isomorphic to the

¢

subspace geometry Σk V . ¤

Proof (a) If v1 vk are linearly independent, then they form part of a basis,

¦ 2§ ¦

§§

and their product is one of the basis vectors of X , hence non-zero. Conversely,

if these vectors are dependent, then one of them can be expressed in terms of the

others, and the product is zero (using linearity and the fact that a product with two

equal terms is zero).

(b) It follows from our remarks about determinants that, if v1 vk are re- ¦ 2§ ¦

§§

placed by another k-tuple with the same span, then v1 vk is multiplied by a ' § '

§§

¢

scalar factor, and the point of PG N 1¦ F it spans is unaltered. If W1 W2 , then ¥

© ¤

we can (as usual in linear algebra) choose a basis for V containing bases for both

W1 and W2 ; the corresponding pure products are distinct basis vectors of X , and

so span distinct points. The correspondence is one-to-one.

Suppose that W1 and W2 are collinear in the subspace geometry. then they have

bases v1 vkQ 1 w1 and v1 vkQ 1 w2 . Then the points spanned by the

¦ 2§ ¦ ¦ ¦ §2§ ¦ ¦

§§ §

vectors

¢

vkQ 1 ±w1 βw2

v1 ' § ' '

§§ £ ¤

¢

form a line in PG N 1¦ F and represent all the points of the line in the subspace

© ¤

geometry joining W1 and W2 .

Conversely, suppose that v1 vk and w1 wk are two pure products. ' 2§ ' ' § '

§§ §§

By factoring out the intersection of the corresponding subspaces, we may assume

that v1 wk are linearly independent. If k 1, then no other vector in the span

¦ § ¦ 5

§§

of these two pure products is a pure product. If k 1, then the three points are ¥

coplanar.

152 10. Exterior powers and Clifford algebras

¢

The other natural geometry on the set Σk V is just the truncation of the pro-

¤

jective geometry to ranks k 1¦ k and k 1; in other words, its varieties are the

© £

subspaces of V of these three ranks, and incidence is inclusion. This geometry has

no immediate connection with exterior algebra; but it (or the more general form

based on any generalised projective geometry) has a beautiful characterisation due

to Sprague (1981).

Theorem 10.3 (a) The geometry just described has diagram

LS L

R R R

¦

where LS denotes the class of dual linear spaces.

(b) Conversely, any geometry with this diagram, in which chains of subspaces are

¬nite, consists of the varieties of ranks k 1¦ k and k 1 of a generalised

© £

projective space of ¬nite dimension, two varieties incident if one contains

the other.

Proof The residue of a variety of rank k 1 is the quotient projective space; and

© ¢

the residue of a variety of rank k 1 is the dual of PG k¦ F . This establishes the

£ ¤

diagram.

I will not give the proof of Sprague™s theorem. the proof is by induction (hence

the need to assume ¬nite rank). Sprague shows that it is possible to recognise

in the geometry objects corresponding to varieties of rank k 2, these objects ©

LS L again, R T

R R

together with the left and centre nodes forming the diagram

but with the dimension of the residue of a variety belonging to the rightmost node

reduced by 1. After ¬nitely many steps, we reach the points, lines and planes of

the projective space, which is recognised by the Veblen“Young axioms.

Exercise

¢

1. Show that the dual of the generalised hexagon G2 F constructed in Sec- ¤ ¢

tion 8.8 is embedded in the subspace geometry of lines of PG 6¦ F . [Hint: the ¤ ¢

lines of the hexagon through a point x are all those containing x in a plane W x .] ¤

10.3 Near polygons

In this section we consider certain special point-line geometries. These ge-

ometries will always be connected, and the distance between two points is the

10.3. Near polygons 153

smallest number of lines in a path joining them. A near polygon is a geometry

with the following property:

(NP) Given any point p and line L, there is a unique point of L nearest to p.

If a near polygon has diameter n, it is called a near 2n-gon.

We begin with some elementary properties of near polygons.

Proposition 10.4 In a near polygon,

(a) two points lie on at most one line;

(b) the shortest circuit has even length.

Proof (a) Suppose that lines L1 L2 contain points p1 p2 . Let q L1 . Then q is at

¦ ¦ ¡

distance 1 from the two points p1 p2 of L2 , and so is at distance 0 from a unique

¦

point of L2 ; that is, q L2 . So L1 L2 ; and, interchanging these two lines, we

¡ E

¬nd that L1 L2 .

¥

If a circuit has odd length 2m 1, then a point lies at distance m from two

£

points of the opposite line; so it lies at distance m 1 from some point of this line,

©

and a circuit of length 2m is formed.

Any generalised polygon is a near polygon; and any “non-degenerate” near

4-gon is a generalised quadrangle (see Exercise 1).

Some deeper structural properties are given in the next two theorems, which

were found by Shult and Yanushka [30].

Theorem 10.5 Suppose that x1 x2 x3 x4 is a circuit of length 4 in a near polygon,

at least one of whose sides contains more than two points. Then there is a unique

subspace containing these four points which is a generalised quadrangle.

A subspace of the type given by this theorem is called a quad.

Corollary 10.6 Suppose that a near polygon has the properties

(a) any line contains more than two points;

(b) any two points at distance 2 are contained in a circuit of length 4.

154 10. Exterior powers and Clifford algebras

Then the points, lines and quads form a geometry belonging to the diagram

L

R R R

§

We now assume that the hypotheses of this Corollary apply. Let p be a point

¢

and Q a quad. We say that the pair p¦ Q is classical if ¤

(a) there is a unique point x of Q nearest p;

¢ ¢

(b) for y Q, d y¦ p d x¦ p 1 if and only if y is collinear with x.

¡ ¥¤ U¤

£

(The point x is the “gateway” to Q from p.) An ovoid in a generalised quadrangle

is a set O of (pairwise non-collinear) points with the property that any further

point of the quadrangle is collinear with a unique point of O. The point-quad pair

¢

p¦ Q is ovoidal if the set of points of Q nearest to p is an ovoid of Q.

¤

Theorem 10.7 In a near polygon with at least three points on a line, any point-

quad pair is either classical or ovoidal.

A proof in the ¬nite case is outlined in Exercise 2.

We now give an example, the sextet geometry of Section 9.3 (which, as we

already know, has the correct diagram). Recall that the POINTs, LINEs, and

“QUADs” (as we will now re-name them) of the geometry are the octads, trios

and sextets of the Witt system. We check that this is a near polygon, and examine

the point-quad pairs.

Two octads intersect in 0, 2 or 4 points. If they are disjoint, they are contained

in a trio (i.e., collinear). If they intersect in four points, they de¬ne a sextet, and

so some octad is disjoint from both; so their distance is 2. If they intersect in two

points, their distance is 3. Suppose that B1 B2 B3 is a trio and B an octad not in

¦ ¦

this trio. Either B is disjoint from (i.e., collinear with) a unique octad in the trio,

or its intersections with them have cardinalities 4, 2, 2. In the latter case, it lies at

distance 2 from one POINT of the LINE, and distance 3 from the other two.

Now let B be a POINT (an octad), and S a QUAD (a sextet). The intersections

of B with the tetrads of S have the property that any two of them sum to 0, 2, 4

or 8; so they are all congruent mod 2. If the intersections have even parity, they

are 4¦ 4¦ 0¦ 0¦ 0¦ 0 (the POINT lies in the QUAD) or 2¦ 2¦ 2¦ 2¦ 0¦ 0 (B is disjoint from

a unique octad incident with S, and the pair is classical). If they have odd parity,

10.4. Dual polar spaces 155

they are 3¦ 1¦ 1¦ 1¦ 1¦ 1; then B has distance 2 from the ¬ve octads containing the¢

¬rst tetrad, and distance 3 from the others. Note that in the GQ of order 2¦ 2 , ¤

represented as the pairs from a 6-set, the ¬ve pairs containing an element of the

¢

6-set form an ovoid. So B¦ S is ovoidal in this case.

¤

10.3.1 Exercises

1. (a) A near polygon with lines of size 2 is a bipartite graph.

(b) A near 4-gon, in which no point is joined to all others, is a generalised

quadrangle.

2. Let Q be a ¬nite GQ with order s¦ t, where s 1. 5

(a) Suppose that the point set of Q is partitioned into three subsets A¦ B¦ C

such that for any line L, the values of L A, L B and L C are either 1¦ s¦ 0,

V VV V V V

F F F

or 0¦ 1¦ s. Prove that A is a singleton, and B the set of points collinear with A.

(b) Suppose that the point set of Q is partitioned into two subsets A and B

such that any line contains a unique point of A. Prove that A is an ovoid.

(c) Hence prove (10.3.4) in the ¬nite case.

10.4 Dual polar spaces

We now look at polar spaces “the other way up”. That is, given an abstract po-

lar space of polar rank n, we consider the geometry whose POINTs and LINEs are

the subspaces of dimension n 1 and n 2 respectively, incidence being reversed

© ©

inclusion. (This geometry was introduced in Section 7.4.)

Proposition 10.8 A dual polar space of rank n is a near 2n-gon.

Proof This is implicit in what we proved in Proposition 7.9.

Any dual polar space has girth 4, and any circuit of length 4 is contained

in a unique quad. Moreover, the point-quad pairs are all classical. Both these

assertions are easily checked in the polar space by factoring out the intersection

of the subspaces in question.

The converse of this result was proved by Cameron [9]. It is stated here using

the notation and ideas (and simpli¬cations) of Shult and Yanushka described in

the last section.

Theorem 10.9 Let be a near 2n-gon. Suppose that

W

156 10. Exterior powers and Clifford algebras

(a) any 4-circuit is contained in a quad;

(b) any point-quad pair is classical;

(c) chains of subspaces are ¬nite.

Then is a dual polar space of rank n.

W

Proof The ideas behind the proof will be sketched.

Given a point p, the residue of p (that is, the geometry of lines and quads

containing p) is a linear space, by hypothesis (a). Using (b), it is possible to show

that this linear space satis¬es the Veblen“Young axioms, and so is a projective

¢

space p (possibly in¬nite-dimensional). We may assume that this geometry

X ¤

has dimension greater than 2 (otherwise the next few steps are vacuous). ¢

Now, given points p and q, let p¦ q be the set of lines through p (i.e., points Y ¤

¢

of p ) which belong to geodesics from p to q (that is, which contain points r

X ¤¢ ¢ ¢

with d q¦ r d p¦ q 1). This set is a subspace of p . Let X be any subspace

¥¤ X

©`¤ ¤

¢

of p , and let

X a

¤ ¢ ¢

p¦ X q : p¦ q X ¥¤ Y b¤

E §

a

¢

It can be shown that p¦ X is a subspace of the geometry, containing all geodesics a

¤ ¢

between any two of its points, and that, if pc is any point of p¦ X , then there is a a ¤

¢ ¢ ¢

a subspace X of pc such that pc X p¦ X . ¤a

c X d

¦ ¥ ¤c

¤ ¢

For the ¬nal step, it is shown that the subspaces p¦ X , ordered by reverse ¤

inclusion, satisfy the axioms (P1)“(P4) of Tits.

Remark In the case when any line has more than two points, condition (a) is a

consequence of (10.3.2), and (10.3.4) shows that (b) is equivalent to the assertion

that no point-quad pairs are ovoidal.

10.5 Clifford algebras and spinors

Spinors provide projective embeddings of some geometries related to dual

polar spaces, much as exterior powers do for subspace geometries. But they are

somewhat elusive, and we have to construct them via Clifford algebras.

Let V be a vector space over a commutative ¬eld F, and f a quadratic form on ¢

V ; let b be the bilinear form obtained by polarising f . The Clifford algebra C f ¤

¢

of f (or of the pair V¦ f ) is the free-est algebra generated by V subject to the

¤

10.5. Clifford algebras and spinors 157

¢

condition that v2 f v 1 for all v V . In other words, it is the quotient of the

¥ ¡

e¤

¢ ¢

tensor algebra T V by the ideal generated by all elements v2 f v 1 for v V . ¡

¤ © ¤

¢

Note that vw wv b v¦ w 1 for v¦ w V . ¥ ¡

£ f¤

The Clifford algebra is a generalisation of the exterior algebra, to which it

reduces if f is identically zero. And it has the same dimension:

¢

Proposition 10.10 Let v1 vn be a basis for V . Then C f has a basis con- ¦ 2§ ¦

§§ ¤ ¢ ¢

2n .

sisting of all vectors vi1 vik , for 0 i1 ik n; and so rk C f ( ( 2§ ¥ ¤

2 §§ ¤

Proof Any product of basis vectors can be rearranged into non-decreasing order,

modulo products of smaller numbers of basis vectors, using

¢

wv vw b v¦ w 1

¥ © f¤ §

A product with two terms equal can have its length reduced. Now the result fol-

lows by multilinearity.

¢

In an important special case, we can describe the structure of C f . ¤

Theorem 10.11 Let f be a split quadratic form of rank n over F (equivalent to

x1 x2 x3 x4 x2nQ 1 x2n

£ hg£

£§§§ §

¢ ¢

M2n F , the algebra of 2n 2n matrices over F.

Then C f

7¥ ¤ ¤

¢

Proof It suf¬ces to ¬nd a linear map θ : V M2n F satisfying

i ¤

¢ ¢

(a) θ V generates M2n F (as algebra with 1);

¤ ¤

¢ ¢

(b) θ v 2 f v I for all v V.

¥ ¡

¤ ¤

¢ ¢

For if so, then M2n F is a homomorphic image of C f ; comparing dimensions, ¤ ¤

they are equal.

We use induction on n. For n 0, the result is trivial. Suppose that it is ¥

¢

true for n, with a map θ. Let V V x¦ y , where f »x µy »µ. De¬ne

˜ ¥ qp

G ¥¤

H £

¢

θ:V

˜˜ M2nr 1 F by

i ¤

¢

θv O

¢

θv

˜ ¤

v V¦

¢

θv

)

¥¤ ¦ ¡

O 1¤

©

158 10. Exterior powers and Clifford algebras

O I OO

¢ ¢

θx θy

˜ ˜

0¥ ¤

) ¦ 0¥ ¤

) ¦

O O IO

1 1

extended linearly. ¢

AB

To show generation, let C D4 M2nr 1 F be given. We may assume induc-

3 ¡ ¤ ¢

tively that A¦ B¦ C¦ D are linear combinations of products of θ v , with v V . The ¡

¤

¢

same combinations of products of θ v have the forms A AO

˜ ˜

O As , etc. Now 3

¥

¤ 4t

A B ¢ ¢ ¢ ¢ ¢ ¢

Aθ x θ y Bθ x θyC θ y Dθ x

˜˜ ˜ ˜˜ ˜ ˜ ˜˜

˜

) ¥ ¤ U¤

£ U¤

£ ¤ £ ¤ u¤

§

C D 1

To establish the relations, we note that

¢

θv »I

¢

θv »x

˜ ¤

µy ¢

θv

0¥ ¤

) ¦

£ £

µI 1¤

©

¢ ¢

»µ IO

and the square of the right-hand side is f v , as required.

3¤ O I4

U¤

£

More generally, the argument shows the following.

Theorem 10.12 If the quadratic form f has rank n and germ f0 , then

¢ ¢ ¢

Cf C f0 M2n F

F

7¥ ¤ ¤ u¤

§

¢

In particular, C x2 x1 x2 x2nQ 1 x2n is the direct sum of two copies of

0 £ v£

£§§§ ¤

¢

M2n F ; and, if ± is a non- square in F, then

¤

¢ ¢

C ±x2 x1 x2 x2nQ 1 x2n M2n K

7¥ ¤ ¦¤

0 £ hg£

£§§§

x¢

w

where K F ± . ¥ ¤

Looked at more abstractly, Theorem 10.12 says that the Clifford algebra of

the split form of rank n is isomorphic to the algebra of endomorphisms of a vector

space S of rank 2n . This space is the spinor space, and its elements are called

spinors. Note that the connection between the spinor space and the original vector

space is somewhat abstract and tenuous! It is the spinor space which carries the

geometrical structures we now investigate.

Exercise

1. Prove that the Clifford algebras of the real quadratic forms x2 and x2 y2 © © ©

respectively are isomorphic to the complex numbers and the quaternions. What is

the Clifford algebra of x2 y2 z2 ? © © ©

10.6. The geometry of spinors 159

10.6 The geometry of spinors

In order to connect spinors to the geometry of the quadratic form, we ¬rst need

to recognise the points of a vector space within its algebra of endomorphisms.

let V be a vector space, A the algebra of linear transformations of V . Then A

is a simple algebra. If U is any subspace of V , then

¢

IU a A : va U for all v V

¥¤ ¡ ¡ ¡

is a left ideal in A. Every left ideal is of this form (see Exercise 1). So the projec-

tive space based on V is isomorphic to the lattice of left ideals of A. In particular,

the minimal left ideals correspond to the points of the projective space. Moreover,

¢

if U has rank 1, then I U has rank n, and A (acting by left multiplication) in-

¤

duces the algebra of linear transformations of U . In this way, the vector space is

“internalised” in the algebra.

Now let V carry a split quadratic form of rank n. If U is a totally singular

subspace of rank n, then the elements of U generate a subalgebra isomorphic to

ˆ

the exterior algebra of U . Let U denote the product of the vectors in a basis of U .

ˆ

Note that U is unchanged, apart from a scalar factor, if a different basis is used.

ˆ

Then vuU 0 whenever v V , u U , and u 0; so the left ideal generated by

¥ ¡ ¡ ¥

ˆ ˆ

U has dimension 2n (with a basis of the form vi1 vik U , where v1 vn is ¦ § ¦

§

§§ §§

ˆ

a basis of a complement for U , and 1 i1 ik n. Thus, U generates a ( ( §

§§

¢

minimal left ideal of C f . By the preceding paragraph, this ideal corresponds to

¤ ¢

a point of the projective space PG 2n 1¦ F based on the spinor space S. © ¤

Summarising, we have a map from the maximal totally singular subspaces of

the hyperbolic quadric to a subset of the points of projective spinor space. The

elements in the image of this map are called pure spinors.

We now state some properties of pure spinors without proof.

Proposition 10.13 (a) There is a decomposition of the spinor space S into two

subspaces S , S , each of rank 2nQ 1 . Any pure spinor is contained in one

% Q

of these subspaces.

(b) Any line of spinor space which contains more than two pure spinors has the

form

U : U is t.s. with rank n¦ U has type µ¦ U W

ˆ G y ¦

H

2, and µ

where W is a t.s. subspace of rank n 1.

¥

©

160 10. Exterior powers and Clifford algebras

In (a), the subspaces S and S are called half-spinor spaces.

% Q

In (b), the type of a maximal t.s. subspace is that described in Section 7.4. The

maximal t.s. subspaces containing W form a dual polar space of rank 2, which in

this case is simply a complete bipartite graph, the parts of the bipartition being

the two types of maximal subspace. Any two subspaces of the same type have

intersection with even codimension at most 2, and hence intersect precisely in W .

The dual polar space associated with the split quadratic form has two points

per line, and so in general is a bipartite graph. The two parts of the bipartition

can be identi¬ed with the pure spinors in the two half-spinor spaces. The lines

described in (b) within each half-spinor space form a geometry, a so-called half-

spinor geometry: two pure spinors are collinear in this geometry if and only if they

lie at distance 2 in the dual polar space. In general, distances in the half-spinor

geometry are those in the dual polar space, halved!

Proposition 10.14 If p is a point and L a line in a half-spinor geometry, then

either there is a unique point of L nearest p, or all points of L are equidistant from

p.

Proof Recall that the line L of the half-spinor geometry is “half” of a complete

bipartite graph Q, which is a quad in the dual polar space. If the gateway to Q is

on L, it is the point of L nearest to p; if it is on the other side, then all points of L

are equidistant from p.

The cases n 3¦ 4 give us yet another way of looking at the Klein quadric and

¥

triality.

Example n 3. The half-spinor space has rank 4. The diameter of the half-

¥

¢

spinor geometry is 1, and so it is a linear space; necessarily PG 3¦ F : that is,¤

every spinor in the half-spinor space is pure. Points of this space correspond to

one family of maximal subspaces on the Klein quadric.

Example n 4. Now the half-spinor spaces have rank 8, the same as V . The

¥

half-spinor space has diameter 2, and (by Proposition 10.14) satis¬es the Buekenhout“

Shult axiom. But we do not need to use the full classi¬cation of polar spaces here,

¢

since the geometry is already embedded in PG 7¦ F ! We conclude that each half-

¤

spinor space is isomorphic to the original hyperbolic quadric.

We conclude by embedding a couple more dual polar spaces in projective

spaces.

10.6. The geometry of spinors 161

Proposition 10.15 Let f be a quadratic form of rank n 1 on a vector space of

©

rank 2n 1. Then the dual polar space of F is embedded as all the points and

©

some of the lines of the half-spinor space associated with a split quadratic form

of rank n.

Proof We can regard the given space as of the form v‚ , where v is a non-singular

vector in a space carrying a split quadratic form of rank n. Now each t.s. subspace

of rank n 1 for the given form is contained in a unique t.s. space of rank n of each

©

type for the split form; so we have an injection from the given dual polar space to

a half-spinor space. The map is onto: for if U is t.s. of rank n, then U c‚ has

F

rank n 1. A line of the dual polar space consists of all the subspaces containing

©

a ¬xed t.s. subspace of rank n 2, and so translates into a line of the half-spinor

©

space, as required.

Proposition 10.16 Let K be a quadratic extension of F, with Galois automor-

phism σ. Let V be a vector space of rank 2n over K, carrying a non-degenerate

σ-Hermitian form b of rank n. Then the dual polar space associated with b is

embeddable in a half-spinor geometry over F.

¢ ¢ ¢

Proof Let H v b v¦ v . Then H v F for all v V ; and H is a quadratic

¥¤ ¡¤ ¡

¤

form on the space VF obtained by restricting scalars to F. (Note that VF has rank

4n over F.) Now any maximal t.i. subspace for b is a maximal t.s. subspace for

H of rank 2n; so H is a split form, and we have an injection from points of the

dual unitary space to pure spinors. Moreover, the intersection of any two of these

maximal t.s. subspaces has even F-codimension in each; so they all have the same

type, and our map goes to points of a half-spinor geometry.

A line of the dual polar space is de¬ned by a t.i. subspace of rank n 1 (over

©

K), which is t.s. of rank 2n 2 over F; so it maps to a line of the half-spinor

©

geometry, as required.

In the case n 3, we have the duality between the unitary and non-split or-

¥

thogonal spaces discussed in Section 8.3.

Exercise

1. (a) Prove that the set of endomorphisms of V with range contained in a

subspace U is a left ideal.

(b) Prove that, if T has range U , then any endomorphism whose range is

contained in U is a left multiple of T .

162 10. Exterior powers and Clifford algebras

(c) Deduce that every left ideal of the endomorphism ring of V is of the

form described in (a).

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Index

abstract polar space, 105 chamber-connected, 70

addition, 1, 10, 32 characteristic, 1

Chevalley group, 133

af¬ne plane, 21, 40

circle, 68

af¬ne space, 3, 6

classical groups, 114, 131

algebraic curve, 52

classical point-quad pair, 154

algebraic variety, 52

classical polar space, 88

alternating bilinear form, 77

Clifford algebra, 156

alternating groups, 131

code, 34

alternative division rings, 114

coding theory, 34

anisotropic, 84

collineation, 8

anti-automorphism, 2

commutative ¬eld, 1, 147

atom, 39

complete bipartite graph, 97

atomic lattice, 39

complete graph, 68

automorphism, 17

con¬guration theorem, 23

axis, 61

conic, 52, 127

Baer subplane, 141 connected geometry, 66

bilinear form, 76 coordinatisation, 9

binary Golay code, 36 corank, 66

bispread, 47, 127 coset geometry, 70

bits, 34 cospread, 47

block, 16 cotype, 66

Buekenhout geometry, 65 cross ratio, 59

buildings, 131 cross-ratio, 8

bundle theorem, 58

degenerate conic, 55

Cayley numbers, 114 derivation, 46

central collineation, 23 Desargues™ Theorem, 4, 22, 38

central collineations, 133 Desargues™ theorem, 133

centre, 61 Desarguesian planes, 24

chamber, 70 Desarguesian spread, 46

168

INDEX 169

design, 16 Galois™ Theorem, 2

determinant, 149 gamma space, 150

diagram, 67 Gaussian coef¬cient, 15

digon, 67 general linear group, 8

dimension, 3 generalised polygons, 131

division ring, 1 generalised projective plane, 68

dodecad, 140, 144 generalised projective space, 39

doubly-even self-dual code, 130 generalised quadrangle, 90, 97

dual polar space, 107, 133 geometry, 65

dual space, 3 germ, 85, 90, 92

duality, 75 ghost node, 142

duality principle, 6 Golay code, 137, 144

GQ, 98

egglike inversive plane, 58 graph

elation, 23, 61 complete bipartite, 97

elliptic quadrics, 57 grid, 91, 98

equianharmonic, 60 group, 8

error-correcting codes, 34 groups of Lie type, 131

exterior algebra, 149

Hadamard matrix, 144

exterior points, 53

exterior power, 148 half-spinor geometry, 160

half-spinor spaces, 160

exterior set, 103

exterior square, 148 Hamming codes, 35

Hamming distance, 34

Feit“Higman theorem, 132 harmonic, 60

¬eld, 1 Hermitian form, 77

¬nite ¬eld, 1, 14 Hessian con¬guration, 141

¬nite simple groups, 131 homology, 23, 63

¬rm, 66 hyperbolic line, 84

¬xed ¬eld, 81 hyperbolic quadric, 91, 103

¬‚ag, 66 hyperoval, 57, 101, 141

¬‚ag geometry, 132 hyperplane, 3, 33, 90

¬‚at, 3 hyperplane at in¬nity, 3, 7

¬‚at C3 -geometry, 102

ideal hyperplane, 7

free plane, 20

incidence relation, 65

Friendship Theorem, 22

interior points, 53

Fundamental Theorem of Projective

Geometry, 8, 114 inversive plane, 58

INDEX

170

isomorphism, 8 orders, 98

orthogonal groups, 114

join, 39 orthogonal space, 88

overall parity check, 137

Kirkman™s schoolgirl problem, 119

ovoid, 57, 58, 125, 131, 135, 154

Klein quadric, 118

ovoidal point-quad pair, 154

lattice, 39

Pappian planes, 25

left vector space, 2

Pappus™ Theorem, 24, 55

Lie algebra, 133

parallel, 7

line, 3, 89, 150

parallel postulate, 21

line at in¬nity, 21

parallelism, 41

linear code, 137

parameters, 69

linear codes, 35

partial linear space, 67

linear diagram, 69

Pascal™s Theorem, 55

linear groups, 131

Pascalian hexagon, 55

linear space, 36, 40, 68

passant, 57

linear transformations, 8

perfect, 83

Mathieu group, 139, 140, 143 perfect codes, 35

matrix, 3, 18 perfect ¬eld, 123

meet, 39 perspectivity, 27

Miquel™s theorem, 58 plane, 3, 89

modular lattice, 39 Playfair™s Axiom, 21, 41

Moufang condition, 114, 133 point, 3, 16, 89

Moulton plane, 20 point at in¬nity, 21

multiplication, 1, 10 point-shadow, 69

multiply transitive group, 11 polar rank, 85, 88

polar space, 88

near polygon, 153

polarisation, 82

Neumaier™s geometry, 101

polarity, 77

non-degenerate, 76

prders, 69

non-singular, 83

prime ¬eld, 1

nucleus, 55

probability, 17

projective plane, 4, 19, 40, 68

octad, 140

projective space, 3

octonions, 114

projectivity, 27

opposite ¬eld, 2

pseudoquadratic form, 82, 113

opposite regulus, 46

order, 19, 22, 58 pure products, 151

INDEX 171

pure spinors, 159 sum of linear spaces, 38

support, 35, 140

quad, 153 Suzuki“Tits ovoids, 58

quadratic form, 82 symmetric algebra, 149

quaternions, 2 symmetric bilinear form, 77

symmetric power, 149

radical, 80, 110

symmetric square, 149

rank, 3, 66, 89

symplectic groups, 114

reduced echelon form, 18

symplectic space, 88

Ree group, 135

symplectic spread, 128

Reed“Muller code, 139

re¬‚exive, 77

t.i. subspace, 88

regular spread, 46, 127

t.s. subspace, 88

regulus, 46, 127

tangent, 57

residually connected geometry, 66

tangent plane, 57

residue, 66

tensor algebra, 148

right vector space, 2

tensor product, 147

ruled quadric, 91

tetrad, 142

Schl¨ ¬‚i con¬guration, 100

a theory of perspective, 7

secant, 57 thick, 37, 66

Segre™s Theorem, 52 thin, 66

semilinear transformations, 8 totally isotropic subspace, 88

semilinear, 76 totally singular subspace, 88

sesquilinear, 76 trace, 81

shadow, 69 trace-valued, 113

sharply t-transitive, 28 trace-valued Hermitian form, 81

singular subspace, 105 transitivity of parallelism, 41

skew ¬eld, 1 translation plane, 45

solid, 41 transvection, 61

solids, 128 transversality condition, 66

spinor space, 158 triality, 129, 133

spinors, 158 triality quadric, 133

sporadic groups, 131 triangle property, 110

spread, 45, 125, 127, 131 trio, 140

Steiner quadruple system, 43 type map, 65

Steiner triple system, 43 types, 65

subspace, 33, 36, 90, 105

subspace geometry, 150 unital, 141

INDEX

172

unitary groups, 114

unitary space, 88

varieties, 65

variety, 16

Veblen™s Axiom, 4, 37, 42

Veblen™s axiom, 32, 40

Wedderburn™s Theorem, 1, 25

weight, 35

Witt index, 85

Witt system, 140