<< стр. 7(всего 7)СОДЕРЖАНИЕ
' ' 2В§  ( ( В§ 
В§В§ В§В§
Вў
its dimension is n . Note that k V 0 if k n rk V .
3 ВҐ 5 ВҐ
&
k4 В¤
The exterior algebra of V is 6
n
#
Вў Вў k
V V
ВҐВ¤ & В¦В¤
k\$ 0

with multiplication deп¬Ѓned as for the tensor algebra. Its rank is в€‘n 0 n 2n . 3 ВҐ
k\$ k4
If Оё is a linear transformation on V , then Оё induces in a natural way linear Вў
transformations k Оё on k V , and k Оё on k V , for all k. If rk V n, then we
" " & & ВҐВ¤
Вў Вў
Оё is a scalar. In fact, Оё det Оё . (This fact is the
n n n
have rk V 1, and so
& ВҐВ¤ & & ВҐ В¤
basis of an abstract, matrix-free, deп¬Ѓnition of the determinant.)

Exercises
1. Let F be a skew п¬Ѓeld, V a right F-vector space, and W a left vector space.
В
Show that it is possible to deп¬Ѓne V F W as an abelian group so that
Вў Вў
В  В  В  В  В  В
v1 v2 w v1 w v2 wВ¦ v w1 w2 v w1 v w2
ВҐ ВҐВ¤
ВЈ В¤ ВЈ ВЈ ВЈ

and Вў Вў
О±w
В  В
vО± w v
ВҐ
В¤ В§ВЁВ¤
2. In the identiп¬Ѓcation of F n F m with the space of n m matrices, show
В

that the rank of a matrix is equal to the minimum r for which the corresponding
tensor can be expressed in the form в€‘r 1 vi wi . Show that, in such a minimal
В
i\$
expression, v1 vr are linearly independent, as are w1 wr .
В¦ В§ В¦ В¦ 2В§ В¦
В§В§ В§В§
3. (a) If K is an extension п¬Ѓeld of F, and n a positive integer, prove that
Вў Вў
В
Mn F K Mn K
F 7ВҐ В¦В¤
В¤
Вў
where Mn F is the ring of n n matrices over F. 
В¤
@В
9
(b) Prove that . 8 8 8CAB8
7ВҐ
4. Deп¬Ѓne 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.
ВҐ
More generally, deп¬Ѓne the kth symmetric power SkV , and calculate its dimension; Вў Вў
and deп¬Ѓne the symmetric algebra S V . If dim V n, show that the symmetric ВҐВ¤
В¤
algebra on V is isomorphic to the polynomial ring in n variables over the base
п¬Ѓeld. Вў Вў
5. Prove that, if Оё is a linear map on V , where rk V n, then n Оё det Оё . ВҐВ¤ ВҐ
& В¤
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 Вў
are a couple of ways of deп¬Ѓning a geometry on the set ОЈk ОЈk V of subspaces ВҐ В¤
Вў Вў
of V of rank k (equivalently, the k 1 -dimensional subspaces of PG n 1В¦ F , 3
which I now describe.
The п¬Ѓrst approach produces a point-line geometry. For each pair U1 U2 of В¦
Вў Вў
subspaces of V with U1 U2 , rk U1В¤ k 1, rk U2 k 1, a line ВҐ ВҐВ¤
Вў 
ОЈk : U1
L U1 U2В¤ W W U2
В¦ ВҐ ВЎ D D В§

Now two points lie in at most one line. For, if W1 W2 are distinct subspaces of В¦
Вў
rank k and W1 W2 L U1 U2 , then U1 W1 W2 and W1 W2 U2 ; so equality
В¦ ВЎ В¦ G В¦
В¤ 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. Вў
In the case k 2, the points of the subspace geometry are the lines of PG n
1В¦ F , and its lines are the plane pencils. In particular, for k 2, n 4, it is the ВҐ ВҐ
В¤
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 п¬Ѓrst. In order to prove the
п¬Ѓrst 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
Вў Вў Вў Вў
where either rk U1В¤ k 1, rk U2 k 2, or rk U1В¤ k 2, rk U2 k 1.
ВҐ ВҐВ¤ ВҐ ВҐВ¤
Вў
So let W1 W2 W3 be pairwise collinear points. If rk W1 W2 W3 k 1,
В¦ В¦ ВҐВ¤
then the three points are contained in a plane of the п¬Ѓrst type; so suppose not.
Вў
Then we have rk W1 W2 W3 k 2; and, by factoring out this intersection, ВҐВ¤
we may assume that k 2. In the projective space, W1 W2 W3 are now three
ВҐ В¦ В¦
pairwise intersecting lines, and so are coplanar. Thus rkG W1 W2 W3 k 1, and В¦ В¦ ВҐH ВЈ
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 BuekenhoutвЂ“Shult sense), and this property has been used in several recent
characterisations (some of which are surveyed by Shult ).
The subspace geometries have natural embeddings in projective spaces given
k
by exterior powers, generalising the Klein quadric. Let X V ; we consider PВҐ
&
Вў n
the projective space PG N 1В¦ F based on X , where N k4 . This projective 3ВҐ
space contains some distinguished points, those spanned by the vectors of the
form v1 vk , for v1 vk V . We call these pure products.
' В§ ' В¦ 2В§ В¦ ВЎ
В§В§ В§В§

Theorem 10.2 (a) v1 vk 0 if and only if v1 vk are linearly depen-
' ' 2В§ ВҐ В¦ В§ В¦
В§В§ В§В§
dent.
Вў
(b) The set of points of PG N 1В¦ F spanned by non-zero pure products, together
with the lines meeting this set in more than two points, is isomorphic to the
Вў
subspace geometry ОЈk V . В¤

Proof (a) If v1 vk are linearly independent, then they form part of a basis,
В¦ 2В§ В¦
В§В§
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- В¦ 2В§ В¦
В§В§
placed by another k-tuple with the same span, then v1 vk is multiplied by a ' В§ '
В§В§
Вў
scalar factor, and the point of PG N 1В¦ F it spans is unaltered. If W1 W2 , then ВҐ
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
В¦ 2В§ В¦ В¦ В¦ В§2В§ В¦ В¦
В§В§  В§ 
vectors
Вў
vkQ 1 О±w1 ОІw2
v1 ' В§ ' '
В§В§ ВЈ В¤
Вў
form a line in PG N 1В¦ 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В§ ' ' В§ '
В§В§ В§В§
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
В¦ В§ В¦ 5
В§В§
of these two pure products is a pure product. If k 1, then the three points are ВҐ
coplanar.
152 10. Exterior powers and Clifford algebras
Вў
The other natural geometry on the set ОЈk V is just the truncation of the pro-
В¤
jective geometry to ranks k 1В¦ 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
В¦

where LS denotes the class of dual linear spaces.
(b) Conversely, any geometry with this diagram, in which chains of subspaces are
п¬Ѓnite, consists of the varieties of ranks k 1В¦ k and k 1 of a generalised
projective space of п¬Ѓnite 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 kВ¦ F . This establishes the
ВЈ В¤
diagram.
I will not give the proof of SpragueвЂ™s theorem. the proof is by induction (hence
the need to assume п¬Ѓnite rank). Sprague shows that it is possible to recognise
in the geometry objects corresponding to varieties of rank k 2, these objects В©
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 п¬Ѓnitely many steps, we reach the points, lines and planes of
the projective space, which is recognised by the VeblenвЂ“Young axioms.

Exercise
Вў
1. Show that the dual of the generalised hexagon G2 F constructed in Sec- В¤ Вў
tion 8.8 is embedded in the subspace geometry of lines of PG 6В¦ F . [Hint: the В¤ Вў
lines of the hexagon through a point x are all those containing x in a plane W x .] В¤

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
В¦ В¦ ВЎ
distance 1 from the two points p1 p2 of L2 , and so is at distance 0 from a unique
В¦
point of L2 ; that is, q L2 . So L1 L2 ; and, interchanging these two lines, we
ВЎ E
п¬Ѓnd that L1 L2 .
ВҐ
If a circuit has odd length 2m 1, then a point lies at distance m from two
ВЈ
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 вЂњnon-degenerateвЂќ 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 .

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
В§

We now assume that the hypotheses of this Corollary apply. Let p be a point
Вў
and Q a quad. We say that the pair pВ¦ Q is classical if В¤

(a) there is a unique point x of Q nearest p;
Вў Вў
(b) for y Q, d yВ¦ p d xВ¦ p 1 if and only if y is collinear with x.
ВЎ ВҐВ¤ UВ¤
ВЈ

(The point x is the вЂњgatewayвЂќ 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
Вў
pВ¦ Q is ovoidal if the set of points of Q nearest to p is an ovoid of Q.
В¤

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 п¬Ѓnite 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
вЂњQUADsвЂќ (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
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 deп¬Ѓne 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
В¦ В¦ 
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 4В¦ 4В¦ 0В¦ 0В¦ 0В¦ 0 (the POINT lies in the QUAD) or 2В¦ 2В¦ 2В¦ 2В¦ 0В¦ 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 3В¦ 1В¦ 1В¦ 1В¦ 1В¦ 1; then B has distance 2 from the п¬Ѓve octads containing theВў
п¬Ѓrst tetrad, and distance 3 from the others. Note that in the GQ of order 2В¦ 2 , В¤
represented as the pairs from a 6-set, the п¬Ѓve pairs containing an element of the
Вў
6-set form an ovoid. So BВ¦ S is ovoidal in this case.
В¤

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
2. Let Q be a п¬Ѓnite GQ with order sВ¦ t, where s 1. 5
(a) Suppose that the point set of Q is partitioned into three subsets AВ¦ BВ¦ C
such that for any line L, the values of L A, L B and L C are either 1В¦ sВ¦ 0,
V VV V V V
F F F
or 0В¦ 1В¦ 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 п¬Ѓnite case.

10.4 Dual polar spaces
We now look at polar spaces вЂњthe other way upвЂќ. 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 . It is stated here using
the notation and ideas (and simpliп¬Ѓcations) 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 п¬Ѓnite.

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 satisп¬Ѓes the VeblenвЂ“Young axioms, and so is a projective
Вў
space p (possibly inп¬Ѓnite-dimensional). We may assume that this geometry
X В¤
has dimension greater than 2 (otherwise the next few steps are vacuous). Вў
Now, given points p and q, let pВ¦ q be the set of lines through p (i.e., points Y В¤
Вў
of p ) which belong to geodesics from p to q (that is, which contain points r
X В¤Вў Вў Вў
with d qВ¦ r d pВ¦ q 1). This set is a subspace of p . Let X be any subspace
ВҐВ¤ X
Вў
of p , and let
X a
В¤ Вў  Вў
pВ¦ X q : pВ¦ q X ВҐВ¤ Y bВ¤
E В§
a
Вў
It can be shown that pВ¦ X is a subspace of the geometry, containing all geodesics a
В¤ Вў
between any two of its points, and that, if pc is any point of pВ¦ X , then there is a a В¤
Вў Вў Вў
a subspace X of pc such that pc X pВ¦ X . В¤a
c X d
В¦ ВҐ В¤c
В¤ Вў
For the п¬Ѓnal step, it is shown that the subspaces pВ¦ X , ordered by reverse В¤
inclusion, satisfy the axioms (P1)вЂ“(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 п¬Ѓeld F, and f a quadratic form on Вў
V ; let b be the bilinear form obtained by polarising f . The Clifford algebra C f В¤
Вў
of f (or of the pair VВ¦ f ) is the free-est algebra generated by V subject to the
В¤
10.5. Clifford algebras and spinors 157
Вў
condition that v2 f v 1 for all v V . In other words, it is the quotient of the
ВҐ ВЎ
eВ¤
Вў Вў
tensor algebra T V by the ideal generated by all elements v2 f v 1 for v V . ВЎ
Вў
Note that vw wv b vВ¦ w 1 for vВ¦ w V . ВҐ ВЎ
ВЈ fВ¤
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:
 Вў
Proposition 10.10 Let v1 vn be a basis for V . Then C f has a basis con- В¦ 2В§ В¦
В§В§  В¤ Вў Вў
2n .
sisting of all vectors vi1 vik , for 0 i1 ik n; and so rk C f  ( ( 2В§  ВҐ В¤
2 В§В§ В¤

Proof Any product of basis vectors can be rearranged into non-decreasing order,
modulo products of smaller numbers of basis vectors, using
Вў
wv vw b vВ¦ w 1
ВҐ В© fВ¤ В§

A product with two terms equal can have its length reduced. Now the result fol-
lows by multilinearity.
Вў
In an important special case, we can describe the structure of C f . В¤

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

x1 x2 x3 x4 x2nQ 1 x2n
ВЈ hgВЈ
ВЈВ§В§В§ В§
Вў Вў
M2n F , the algebra of 2n 2n matrices over F.
Then C f 
7ВҐ В¤ В¤
Вў
Proof It sufп¬Ѓces to п¬Ѓnd a linear map Оё : V M2n F satisfying
i В¤
Вў Вў
(a) Оё V generates M2n F (as algebra with 1);
В¤ В¤
Вў Вў
(b) Оё v 2 f v I for all v V.
ВҐ ВЎ
В¤ В¤
Вў Вў
For if so, then M2n F is a homomorphic image of C f ; comparing dimensions, В¤ В¤
they are equal.
We use induction on n. For n 0, the result is trivial. Suppose that it is ВҐ
Вў
true for n, with a map Оё. Let V V xВ¦ y , where f О»x Вµy О»Вµ. Deп¬Ѓne
Лњ ВҐ qp
G ВҐВ¤
H ВЈ
Вў
Оё:V
ЛњЛњ M2nr 1 F by
i В¤
Вў
Оёv O
Вў
Оёv
Лњ В¤
v VВ¦
Вў
Оёv
)
ВҐВ¤ В¦ ВЎ
O 1В¤
158 10. Exterior powers and Clifford algebras

O I OO
Вў Вў
Оёx Оёy
Лњ Лњ
0ВҐ В¤
) В¦ 0ВҐ В¤
) В¦
O O IO
1 1
extended linearly. Вў
AB
To show generation, let C D4 M2nr 1 F be given. We may assume induc-
3 ВЎ В¤ Вў
tively that AВ¦ BВ¦ CВ¦ D are linear combinations of products of Оё v , with v V . The ВЎ
В¤
Вў
same combinations of products of Оё v have the forms A AO
Лњ Лњ
O As , etc. Now 3
ВҐ
В¤ 4t
A B Вў Вў Вў Вў Вў Вў
AОё x Оё y BОё x ОёyC Оё y DОё x
ЛњЛњ Лњ ЛњЛњ Лњ Лњ ЛњЛњ
Лњ
) ВҐ В¤ UВ¤
ВЈ UВ¤
ВЈ В¤ ВЈ В¤ uВ¤
В§
C D 1

To establish the relations, we note that
Вў
Оёv О»I
Вў
Оёv О»x
Лњ В¤
Вµy Вў
Оёv
0ВҐ В¤
) В¦
ВЈ ВЈ
ВµI 1В¤
Вў Вў
О»Вµ IO
and the square of the right-hand side is f v , as required.
3В¤ O I4
UВ¤
ВЈ

More generally, the argument shows the following.
Theorem 10.12 If the quadratic form f has rank n and germ f0 , then
Вў Вў Вў
В
Cf C f0 M2n F
F
7ВҐ В¤ В¤ uВ¤
В§

Вў
In particular, C x2 x1 x2 x2nQ 1 x2n is the direct sum of two copies of
0 ВЈ vВЈ
ВЈВ§В§В§ В¤
Вў
M2n F ; and, if О± is a non- square in F, then
В¤
Вў Вў
C О±x2 x1 x2 x2nQ 1 x2n M2n K
7ВҐ В¤ В¦В¤
0 ВЈ hgВЈ
ВЈВ§В§В§
xВў
w
where K F О± . ВҐ В¤
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 В© В© В©
respectively are isomorphic to the complex numbers and the quaternions. What is
the Clifford algebra of x2 y2 z2 ? В© В© В©
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 п¬Ѓrst 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
Вў 
IU a A : va U for all v V
ВҐВ¤ ВЎ ВЎ ВЎ 

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,
Вў
if U has rank 1, then I U has rank n, and A (acting by left multiplication) in-
В¤
duces the algebra of linear transformations of U . In this way, the vector space is
вЂњinternalisedвЂќ 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
Л†
the exterior algebra of U . Let U denote the product of the vectors in a basis of U .
Л†
Note that U is unchanged, apart from a scalar factor, if a different basis is used.
Л†
Then vuU 0 whenever v V , u U , and u 0; so the left ideal generated by
ВҐ ВЎ ВЎ ВҐ
 
Л† Л†
U has dimension 2n (with a basis of the form vi1 vik U , where v1 vn is В¦ В§ В¦
В§
В§В§  В§В§ 
Л†
a basis of a complement for U , and 1 i1 ik n. Thus, U generates a  ( ( В§ 
В§В§
Вў
minimal left ideal of C f . By the preceding paragraph, this ideal corresponds to
В¤ Вў
a point of the projective space PG 2n 1В¦ F based on the spinor space S. В© В¤
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 nВ¦ U has type ОµВ¦ U W
Л† G y В¦
H

2, and Оµ
where W is a t.s. subspace of rank n 1.
ВЃВҐ
ВЂ
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 identiп¬Ѓed 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 вЂњhalfвЂќ 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 3В¦ 4 give us yet another way of looking at the Klein quadric and
ВҐ
triality.

Example n 3. The half-spinor space has rank 4. The diameter of the half-
ВҐ
Вў
spinor geometry is 1, and so it is a linear space; necessarily PG 3В¦ F : that is,В¤
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
ВҐ
half-spinor space has diameter 2, and (by Proposition 10.14) satisп¬Ѓes the BuekenhoutвЂ“
Shult axiom. But we do not need to use the full classiп¬Ѓcation of polar spaces here,
Вў
since the geometry is already embedded in PG 7В¦ F ! We conclude that each half-
В¤
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 vВ‚ , 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 cВ‚ has
F
rank n 1. A line of the dual polar space consists of all the subspaces containing
a п¬Ѓxed 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 Пѓ. Let V be a vector space of rank 2n over K, carrying a non-degenerate
Пѓ-Hermitian form b of rank n. Then the dual polar space associated with b is
embeddable in a half-spinor geometry over F.
Вў Вў Вў
Proof Let H v b vВ¦ v . Then H v F for all v V ; and H is a quadratic
ВҐВ¤ ВЎВ¤ ВЎ
В¤
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 deп¬Ѓned 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-
ВҐ
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
afп¬Ѓne plane, 21, 40
circle, 68
afп¬Ѓne 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 п¬Ѓeld, 1, 147
atom, 39
complete bipartite graph, 97
atomic lattice, 39
complete graph, 68
automorphism, 17
conп¬Ѓguration 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 DesarguesвЂ™ Theorem, 4, 22, 38
central collineations, 133 DesarguesвЂ™ theorem, 133
centre, 61 Desarguesian planes, 24
chamber, 70 Desarguesian spread, 46

168
INDEX 169

design, 16 GaloisвЂ™ Theorem, 2
determinant, 149 gamma space, 150
diagram, 67 Gaussian coefп¬Ѓcient, 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
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
FeitвЂ“Higman theorem, 132 harmonic, 60
п¬Ѓeld, 1 Hermitian form, 77
п¬Ѓnite п¬Ѓeld, 1, 14 Hessian conп¬Ѓguration, 141
п¬Ѓnite simple groups, 131 homology, 23, 63
п¬Ѓrm, 66 hyperbolic line, 84
п¬Ѓxed п¬Ѓeld, 81 hyperbolic quadric, 91, 103
п¬‚ag, 66 hyperoval, 57, 101, 141
п¬‚ag geometry, 132 hyperplane, 3, 33, 90
п¬‚at, 3 hyperplane at inп¬Ѓnity, 3, 7
п¬‚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
KirkmanвЂ™s schoolgirl problem, 119
ovoid, 57, 58, 125, 131, 135, 154
ovoidal point-quad pair, 154
lattice, 39
Pappian planes, 25
left vector space, 2
PappusвЂ™ Theorem, 24, 55
Lie algebra, 133
parallel, 7
line, 3, 89, 150
parallel postulate, 21
line at inп¬Ѓnity, 21
parallelism, 41
linear code, 137
parameters, 69
linear codes, 35
partial linear space, 67
linear diagram, 69
PascalвЂ™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 п¬Ѓeld, 123
meet, 39 perspectivity, 27
MiquelвЂ™s theorem, 58 plane, 3, 89
modular lattice, 39 PlayfairвЂ™s Axiom, 21, 41
Moufang condition, 114, 133 point, 3, 16, 89
Moulton plane, 20 point at inп¬Ѓnity, 21
multiplication, 1, 10 point-shadow, 69
multiply transitive group, 11 polar rank, 85, 88
polar space, 88
near polygon, 153
polarisation, 82
NeumaierвЂ™s geometry, 101
polarity, 77
non-degenerate, 76
prders, 69
non-singular, 83
prime п¬Ѓeld, 1
nucleus, 55
probability, 17
projective plane, 4, 19, 40, 68
projective space, 3
octonions, 114
projectivity, 27
opposite п¬Ѓeld, 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 SuzukiвЂ“Tits ovoids, 58
quadratic form, 82 symmetric algebra, 149
quaternions, 2 symmetric bilinear form, 77
symmetric power, 149
symmetric square, 149
rank, 3, 66, 89
symplectic groups, 114
reduced echelon form, 18
symplectic space, 88
Ree group, 135
ReedвЂ“Muller code, 139
reп¬‚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
SchlВЁ п¬‚i conп¬Ѓguration, 100
a theory of perspective, 7
secant, 57 thick, 37, 66
SegreвЂ™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 п¬Ѓeld, 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
VeblenвЂ™s Axiom, 4, 37, 42
VeblenвЂ™s axiom, 32, 40

WedderburnвЂ™s Theorem, 1, 25
weight, 35
Witt index, 85
Witt system, 140

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