<<

. 6
( 7)



>>

(a) Show that the map
X Ker X Im X
P fhgQ ¤ 
  
induces a bijection between the point set of the quadric and P P, where
I 
P is the projective line over F.

(b) If A is a non-singular matrix, show that
W
A X : Ker X A Im X
X PR( Q I ¤)
A

£  £ 
W
which corresponds under this bijection to the set p p A :p P) .
(  
¤

(c) Show that, if π is any permutation of P, then pπ p :p
W
P) is a special
( S

¤
 
set; and all special sets have this form.

(d) Deduce that every special set is a quadric if and only if F 3.
1 61

8.4. Dualities of symplectic quadrangles 123

3. Consider the case n 3. Take to be the Klein quadric. Show that the
I
£
Klein correspondence maps the special set to a set S of lines of PG 3 F with 
¤
d 
the property that the set of lines of S through any point of p, or the set of lines
of S in any plane Π, is a plane pencil. Show that the correspondence p Π of fh
PG 3 F , where the set of lines of S containing p and the set contained in Π are

¤
equal, is a symlectic polarity with S as its set of absolute lines. Deduce that S is
the set of lines of a symplectic GQ in PG 3 F , and hence that is a quadric. 
¤
 d
4. Prove by induction on n that, for n 3, any special set is a quadric. (See i
Cameron and Kantor [12] for a crib.)


8.4 Dualities of symplectic quadrangles
A ¬eld of characteristic 2 is said to be perfect if every element is a square.
A ¬nite ¬eld of characteristic 2 is perfect, since the multiplicative group has odd
order.
If F has characteristic 2, then the map x x2 is a homomorphism, since fh
2
x2 y2
x y
   ¤
 £
2
x2 y2
xy  ¤
 £
and is one-to-one. Hence F is perfect if and only if this map is an automorphism.

Theorem 8.5 Let F be a perfect ¬eld of characteristic 2. Then there is an isomor-
phism between the symplectic polar space of rank n over F, and the orthogonal
polar space of rank n de¬ned by a quadratic form in 2n 1 variables. 

Proof Let V be a vector space of rank 2n 1 carrying a non-singular quadratic 
form f of rank n. By polarising f , we get an alternating bilinear form b, which
cannot be non-degenerate; its radical R V is of rank 1, and the restriction of f X
£
to it is the germ of f .
Let W0 be a totally singular subspace of V . Then W W0 R is a totally 
£
isotropic subspace of the non-degenerate symplectic space V R. So we have an &
incidence-preserving injection θ : W0 W0 R R from the orthogonal polar
fh &

space to the symplectic. We have to show that θ is onto.


So let W R be t.i. This means that W itself is t.i. for the form b; but R W , so
& p
W is not t.s. for f . However, on W , we have
f w1 w2 f w1 f w2
  R
 ¤
 £  
f ±w ±2 f w
 ¤
 £ 
124 8. The Klein quadric and triality

so f is semilinear on W . Thus, the kernel of f is a hyperplane W0 of W . The space
W0 is t.s., and W0 R W ; so W0 maps onto W R under θ. &

£
Now consider the case n 2. We have an isomorphism between the symplectic
£
and orthogonal quadrangles, by Theorem 8.5, and a duality, by Theorem 8.3. So:

Theorem 8.6 The symplectic generalised quadrangle over a perfect ¬eld of char-
acteristic 2 is self-dual.

When is there a polarity?

Theorem 8.7 Let F be a perfect ¬eld of characteristic 2. Then the symplectic GQ
over F has a polarity if and only if F has an automorphism σ satisfying

σ2 2 ¤
£
x2 .
where 2 denotes the automorphism x fh

Proof For this, we cannot avoid using coordinates! We take the vector space F 4
with the standard symplectic form

b x1 x2 x3 x4 y1 y2 y3 y4 x1 y2 x2 y1 x3 y4 x4 y3
    
¤ ¤ ¤ ¤ ¤ ¤ ¤
 £
(Remember that the characteristic is 2.) The Klein correspondence takes the line
spanned by x1 x2 x3 x4 and y1 y2 y3 y4 to the point with coordinates zi j , 1 
 
¤ ¤ ¤ ¤ ¤ ¤
 
i j 4, where zi j xi y j x j yi ; this point lies on the quadric with equation
  
£
z12 z34 z13 z24 z14 z23 0
  ¤
£
and (if the line is t.i.) also on the hyperplane z12 z34 0. If we factor out 
£
the subspace spanned by the point with z12 z34 1, zi j 0 otherwise, and use
£ £ £
coordinates z13 z24 z14 z23 , we obtain a point of the symplectic space; the map
¤ ¤ ¤
δ from lines to points is the duality previously de¬ned.


To compute the image of a point p a1 a2 a3 a4 under the duality, take two 
¤ ¤ ¤


t.i. lines through this point and calculate their images. If a1 and a4 are non-zero,
we can use the lines joining p to the points a1 a2 0 0 and 0 a4 a1 0 ; the im- ¤ ¤
¤ ¤ ¤ ¤
 
ages are a1 a3 a2 a4 a1 a4 a2 a3 and a2 a2 0 a1a2 a3 a4 . Now the image of the
14   
¤ ¤ ¤ ¤ ¤ ¤
 
line joining these points is found to be the point a2 a2 a2 a2 . The same formula
1234 
¤ ¤ ¤
is found in all cases. So δ

2 is the collineation induced by the ¬eld automorphism

x x2 , or 2 as we have called it.
fh
8.4. Dualities of symplectic quadrangles 125

Suppose that there is a ¬eld automorphism σ with σ2 2, and let θ σ 1 ; !
then δθ 2 is the identity, so δθ is a polarity.
£ £


Conversely, suppose that there is a polarity. By Theorem 7.14, any collineation
g is induced by the product of a linear transformation and a uniquely de¬ned ¬eld
automorphism θ g . Now any duality has the form δg for some collineation g;


and
θ δg 2 2θ g 2 
 
 £ 
So, if δg is a polarity, then 2θ g 1, whence σ θg satis¬es σ2
2 1 2.
!
 
 £ £  £
In the case where F is a ¬nite ¬eld GF 2m , the automorphism group of F is 
cyclic of order m, generated by 2; and so there is a solution of σ2 2 if and only if

£
m is odd. We conclude that the symplectic quadrangle over GF 2m has a polarity 

if and only if m is odd.
We now examine the set of absolute points and lines (i.e., those incident with
their image). A spread is a set S of lines such that every point lies on a unique line
of S. Dually, an ovoid in a GQ is a set O of points with the property that any line
contains a unique point of O. Note that this is quite different from the de¬nition
of an ovoid in PG 3 F given in Section 4.4; but there is a connection, as we will

¤

see.

Proposition 8.8 The set of absolute points of a polarity of a GQ is an ovoid, and
the set of absolute lines is a spread.

Proof Let δ be a polarity. No two absolute points are collinear. For, if x and y
are absolute points lying on the line L, then x y and Lδ would form a triangle. ¤
Suppose that the line L contains no absolute point. Then L is not absolute, so
W W
Lδ L. Thus, there is a unique line M containing Lδ and meeting L. Then Mδ L,
V
so Mδ is not absolute. But L meets M, so Lδ and Mδ are collinear; hence Lδ Mδ ¤
and L M form a triangle.
H
The second statement is dual.

Theorem 8.9 The set of absolute points of a polarity of a symplectic GQ in
PG 3 F is an ovoid in PG 3 F .
 
¤ ¤

Proof Let σ be the polarity of the GQ , and the polarity of the projective E r
space de¬ning the GQ. By the last result, the set of absolute points of σ is s
an ovoid in . This means that the t.i. lines are tangents to , and the t.i. lines
E s
through a point of form a plane pencil. So we have to prove that any other line
s
of the projective space meets in 0 or 2 points. s
126 8. The Klein quadric and triality

, and pσ L. Then L meets the
Let X be a hyperbolic line, p a point of X cH
s
hyperbolic line LX in a point q. Let qσ M. Since q L, we have p M; so M
£
W W

also meets X , in a point r. Let N rσ . Then q N, so N meets X . Also, N meets
£ W
X
in a point s. The line sσ contains s and N σ r. So s is on two lines meeting
£
s
£
W
X , whence s X . So, if X 1, then X 2. 1 6DscH
i1 1 i61DscH
X
Now let pa be another point of X , and de¬ne La and qa as before. Let K tH
s
be the line pqa . Then p K, so pσ L contains x K σ . Also, K meets La , so
W
£ £
x is collinear with pa . But the only point of L collinear with pa is q. So x q,
£
independent of pa . This means that there is only one point pa p in X , and u
V `H
s
£
this set has cardinality 2.



Remark Over ¬nite ¬elds, any ovoid in a symplectic GQ is an ovoid in the
ambient projective 3-space. This is false for in¬nite ¬elds. (See Exercises 2 and
3.)
Hence, if F is a perfect ¬eld of characteristic 2 in which σ2 2 for some
automorphism σ, then PG 3 F possesses symplectic ovoids and spreads. These
£

¤

give rise to inversive planes and to translation planes, as described in Sections 4.1
and 4.4. For ¬nite ¬elds F, these are the only known ovoids other than elliptic
quadrics.



Exercises

1. Suppose that the points and lines of a GQ are all the points and some of the
lines of PG 3 F . Prove that the lines through any point form a plane pencil, and

¤

deduce that the GQ is symplectic.
2. Prove that an ovoid in a symplectic GQ over the ¬nite ¬eld GF q is s 

an ovoid in PG 3 q . [Hint: as in Theorem 8.3.5, it suf¬ces to prove that any


hyperbolic line meets in 0 or 2 points. Now, if X is a hyperbolic line with
s
/ /
0, then X 0, so at most half of the q2 q2 1 hyperbolic lines
X qvH
Vs vH X
s  
12 2
£ £ 
meet . Take any N 2 q q 1 hyperbolic lines including all those meeting ,
s s
 
and let ni of the chosen lines meet in i points. Prove that ‘ ni N, ‘ ini 2N,
£ 
s
‘ i i 1 ni 2N.]
£ £
¡ 
 £
3. Prove that, for any in¬nite ¬eld F, there is an ovoid of the symplectic
quadrangle over F which is not an ovoid of the embedding projective space.
8.5. Reguli and spreads 127

8.5 Reguli and spreads
We met in Section 4.1 the concepts of a regulus in PG 3 F (the set of common

¤

transversals to three pairwise skew lines), a spread (a set of pairwise skew lines
covering all the points), a bispread (a spread containing a line of each plane), and
a regular spread (a spread containing the regulus through any three of its lines).
We now translate these concepts to the Klein quadric.

Theorem 8.10 Under the Klein correspondence,
(a) a regulus corresponds to a conic, the intersection if with a non-singular
I
plane Π, and the opposite regulus to the intersection of with ΠX ; I

(b) a bispread corresponds to an ovoid, a set of pairwise non-perpendicular
points meeting every plane on ; I

(c) a regular spread corresponds to the ovoid W , where W is a line disjoint
vI
H X
from .
I


Proof (a) Take three pairwise skew lines. They translate into three pairwise non-
perpendicular points of , which span a non-singular plane Π (so that Π is
I wI
H
a conic C). Now ΠX is also a non-singular plane, and ΠX is a conic Ca , con-
xI
H
sisting of all points perpendicular to the three given points. Translating back, Ca
corresponds to the set of common transversals to the three given lines. This set is
a regulus, and is opposite to the regulus spanned by the given lines (corresponding
to C).
(b) This is straightforward translation. Note, incidentally, that a spread (or a
cospread) corresponds to what might be called a “semi-ovoid”, were it not that this
term is used for a different concept: that is, a set of pairwise non-perpendicular
points meeting every plane in one family on . I
(c) A regular spread is “generated” by any four lines not contained in a regulus,
in the sense that it is obtained by repeatedly adjoining all the lines in a regulus
through three of its lines. On , the four given lines translate into four points, and
I
the operation of generation leaves us within the 3-space they span. This 3-space
has the form W for some line W ; and no point of can be perpendicular to
X I
every point of such a 3-space.

Note that a line disjoint from is anisotropic; such lines exist if and only if
I
there is an irreducible quadratic over F, that is, F is not quadratically closed. (We
128 8. The Klein quadric and triality

saw earlier the construction of regular spreads: if K is a quadratic extension of F,
take the rank 1 subspaces of a rank 2 vector space over K, and restrict scalars to
F.)
Thus a bispread is regular if and only if the corresponding ovoid is contained
in a 3-space section of . A bispread whose ovoid lies in a 4-space section of
I
is called symplectic, since its lines are totally isotropic with respect to some
I
symplectic form (by the results of Section 8.3). An open problem is to ¬nd a
simple structural test for symplectic bispreads (resembling the characterisation of
regular spreads in terms of reguli).
We also saw in Section 4.1 that spreads of lines in projective space give rise
to translation planes; and regular spreads give Desarguesian (or Pappian) planes.
Another open problem is to characterise the translation planes arising from sym-
plectic spreads or bispreads.


8.6 Triality
Now we increase the rank by 1, and let be a hyperbolic quadric in PG 7 F ,
I 
¤

de¬ned by a quadratic form of rank 4. The maximal t.s. subspaces have dimension
3, and are called solids; as usual, they fall into two families 1 and 2 , so that y y
two solids in the same family meet in a line or are disjoint, while two solids in
different families meet in a plane or a point. Any t.s. plane lies in a unique solid
of each type. Let and be the sets of points and lines.
€ 
Consider the geometry de¬ned as follows.
F
The POINTs are the elements of 1.
y
F
The LINEs are the elements of . 
W W
F
The PLANEs are incident pairs p M , p ,M 2.
€ y

¤

F
The SOLIDs are the elements of 2.
„ƒ€
y‚

Incidence is de¬ned as follows. Between POINTs, LINEs and SOLIDs, it is as
in the quadric, with the additional rule that the POINT M1 and SOLID M2 are
incident if they intersect in a plane. The PLANE p M is incident with all those 
¤

varieties incident with both p and M.

Proposition 8.11 The geometry just described is an abstract polar space in which
any PLANE is incident with just two SOLIDs.
8.7. An example 129

Proof We consider the axioms in turn.
W
(P1): Consider, for example, the SOLID M 2 . The POINTs incident with y
M are bijective with the planes of M; the LINEs are the lines of M; the PLANEs
W
are pairs p M with p M, and so are bijective with the points of M. Incidence is

¤

de¬ned so as to make the subspaces contained in M a projective space isomorphic
to the dual of M.
W
For the SOLID p , the argument is a little more delicate. The geometry
€
pX p is a hyperbolic quadric in PG 5 F , that is, the Klein quadric; the POINTs,

& 
¤

LINEs and PLANEs incident with p are bijective with one family of planes, the
lines, and the other family of planes on the quadric; and hence (by the Klein
correspondence) with the points, lines and planes of PG 3 F . 
¤

The other cases are easier.
W
(P2) is trivial, (P3) routine, and (P4) is proved by observing that if p and €
W
M 2 are not incident, then no POINT can be incident with both.
y
Finally, the SOLIDs containing the PLANE p M are p and M only. 
¤

So the new geometry we constructed is itself a hyperbolic quadric in PG 7 F , 
¤
and hence isomorphic to the original one. This implies the existence of a map „


which carries to itself and . This map is called a triality of
1 2
 ‡h†€
y y‡h ˆh
€
the quadric, by analogy with dualities of projective spaces.
It is more dif¬cult to describe trialities in coordinates. An algebraic approach
must wait until Chapter 10.

Exercise
1. Prove the Buekenhout-Shult property for the geometry constructed in this
W W
section. That is, let M 1, L , and suppose that L is not incident with M;
y 
prove that either all members of 1 containing L meet M in a plane, or just one
y
does, depending on whether L is disjoint from M or not.


8.7 An example
In this section we apply triality to the solution of a combinatorial problem ¬rst
posed and settled by Breach and Street [2]. Our approach follows Cameron and
Praeger [13].
Consider the set of planes of AG 3 2 . They form a 3- 8 4 1 design, that is, ¤ ¤
¤
 
a collection of fourteen 4-subsets of an 8-set, any three points contained in exactly
one of them. There are 8 70 4-subsets altogether; can they be partitioned into
‰
4 £
130 8. The Klein quadric and triality

¬ve copies of AG 3 2 ? The answer is “no”, as has been known since the time of


Cayley. (In fact, there cannot be more than two disjoint copies of AG 3 2 on an ¤

8-set; a construction will be given in the next chapter.) Breach and Street asked:
what if we take a 9-set? This has 9 126 4-subsets, and can conceivably be ‰
4 £
partitioned into nine copies of AG 3 2 , each omitting one point. They proved: ¤


Theorem 8.12 There are exactly two non-isomorphic ways to partition the 4-
subsets of a 9-set into nine copies of AG 3 2 . Both admit 2-transitive groups. ¤



Proof First we construct the two examples.
1. Regard the 9-set as the projective line over GF 8 . If any point is designated 

as the point at in¬nity, the remaining points form an af¬ne line over GF 8 , and 

hence (by restricting scalars) an af¬ne 3-space over GF 2 . We take the fourteen 

planes of this af¬ne 3-space as one of our designs, and perform the same con-
struction for each point to obtain the desired partition. This partition is invariant
under the group P“L 2 8 , of order 9 8 7 3 1512. The automorphism group
¤   
 £
is the stabiliser of the object in the symmetric group; so the number of partitions
isomorphic to this one is the index of this group in S9 , which is 9!& 1512 240.
£
2. Alternatively, the nine points carry the structure of af¬ne plane over GF 3 . 

Identifying one point as the origin, the structure is a rank 2 vector space over
GF 3 . Put a symplectic form b on the vector space. Now there are six 4-sets


which are symmetric differences of two lines through the origin, and eight 4-sets
of the form v) w:b v w 1) for non-zero v. It is readily checked that
( c‘
(‚ 
¤ £
these fourteen sets form a 3-design. Perform this construction with each point
designated as the origin to obtain a partition. This one is invariant under the
group ASL 2 3 generated by the translations and Sp 2 3 SL 2 3 , of order
¤ ¤ ¤
  £ 
9 8 3 216, and there are 9!& 216 1680 partitions isomorphic to this one.
 
£ £
Now we show that there are no others. We use the terminology of coding the-
ory. Note that the fourteen words of weight 4 supporting planes of AG 3 2 , to- ¤

gether with the all-0 and all-1 words, form the extended Hamming code of length
8 (the code we met in Section 3.2, extended by an overall parity check); it is the
only doubly-even self-dual code of length 8 (that is, the only code C CX with
£
all weights divisible by 4).
Let V be the vector space of all words of length 9 and even weight. The
1
function f v 2 wt v mod 2 is a quadratic form on V , which polarises to
  
 £  
the usual dot product. Thus maximal t.s. subspaces for f are just doubly even
self-dual codes, and their existence shows that f has rank 4 and so is the split
8.8. Generalised polygons 131

form de¬ning the triality quadric. (The quadric consists of the words of weight
I
4 and 8.)
Suppose we have a partition of the 4-sets into nine af¬ne spaces. An easy
counting argument shows that every point is excluded by just one of the designs.
So if we associate with each design the word of weight 8 whose support is its
point set, we obtain a solid on the quadric, and indeed a spread or partition of the
quadric into solids.
All these solids belong to the same family, since they are pairwise disjoint. So
we can apply the triality map and obtain a set of nine points which are pairwise
non-collinear, that is, an ovoid. Conversely, any ovoid gives a spread. In fact, an
ovoid gives a spread of solids of each family, by applying triality and its inverse.
So the total number of spreads is twice the number of ovoids.
The nine words of weight 8 form an ovoid. Any ovoid is equivalent to this
one. (Consider the Gram matrix of inner products of the vectors of an ovoid; this
must have zeros on the diagonal and ones elsewhere.) The stabiliser of this ovoid
is the symmetric group S9 . So the number of ovoids is the index of S9 in the
orthogonal group, which turns out to be 960. Thus, the total number of spreads is
1920 240 1680, and we have them all!

£

8.8 Generalised polygons
Projective and polar spaces are important members of a larger class of geome-
tries called buildings. Much of the importance of these derives from the fact that
they are the “natural” geometries for arbitrary groups of Lie type, just as projective
spaces are for linear groups and polar spaces for classical groups. The groups of
Lie type include, in particular, all the non-abelian ¬nite simple groups except for
the alternating groups and the twenty-six sporadic groups. I do not intend to dis-
cuss buildings here ” for this, see the lecture notes of Tits [S] or the recent books
by Brown [C] and Ronan [P] ” but will consider the rank 2 buildings, or gener-
alised polygons as they are commonly known. These include the 2-dimensional
projective and polar spaces (that is, projective planes and generalised quadran-
gles).
Recall that a rank 2 geometry has two types of varieties, with a symmetric
incidence relation; it can be thought of as a bipartite graph. We use graph-theoretic
terminology in the following de¬nition. A rank 2 geometry is a generalised n-gon
(where n 2) if
i

(GP1) it is connected with diameter n and girth 2n;
132 8. The Klein quadric and triality

(GP2) for any variety x, there is a variety y at distance n from x.

It is left to the reader to check that, for n 2 3 4, this de¬nition coincides ¤ ¤
£
with that of a digon, generalised projective plane or generalised quadrangle re-
spectively.
Let be a generalised n-gon. The ¬‚ag geometry of has as POINTs the
E E
varieties of (of both types), and as LINEs the ¬‚ags of , with the obvious
E E
incidence between POINTs and LINEs. It is easily checked to be a generalised
2n-gon in which every line has two points; and any generalised 2n-gon with two
points per line is the ¬‚ag geometry of a generalised n-gon. In future, we usually
assume that our polygons are thick, that is, have at least three varieties of one
type incident with each variety of the other type. It is also easy to show that a
thick generalised polygon has orders, that is, the number of points per line and
the number of lines per point are both constant; and, if n is odd, then these two
constants are equal. [Hint: in general, if varieties x and y have distance n, then
each variety incident with x has distance n 2 from a unique variety incident with
¡
y, and vice versa.]
We let s 1 and t 1 denote the numbers of points per line or lines per point,
 
respectively, with the proviso that either or both may be in¬nite. (If both are ¬nite,
then the geometry is ¬nite.) The geometry is thick if and only if s t 1. The major ’
¤
theorem about ¬nite generalised polygons is the Feit“Higman theorem (Feit and
Higman [17]:

Theorem 8.13 A thick generalised n-gon can exist only for n 2 3 4 6 or 8.
¤ ¤ ¤
£

In the course of the proof, Feit and Higman derive additional information:
F
if n 6, then st is a square;
£
F
if n 8, then 2st is a square.
£
Subsequently, further numerical restrictions have been discovered; for exam-
ple:

s2 and s t 2;
F
if n 4 or n 8, then t  
£ £
s3 and s t 3.
F
if n 6, then t  
£
In contrast to the situation for n 3 and n 4, the only known ¬nite thick
£ £
generalised 6-gons and 8-gons arise from groups of Lie type. There are 6-gons
8.9. Some generalised hexagons 133

with s t q and with s q, t q3 for any prime power q; and 8-gons with
£ £ £ £
s q, t q2 , where q is an odd power of 2. In the next section, we discuss a class
£ £
of 6-gons including the ¬rst-mentioned ¬nite examples.
There is no hope of classifying in¬nite generalised n-gons, which exist for all
n (Exercise 2). However, assuming a symmetry condition, the Moufang condition,
which generalises the existence of central collineations in projective planes, and
is also equivalent to a generalisation of Desargues™ theorem, Tits [35, 36] and
Weiss [39] derived the same conclusion as Feit and Higman, namely, that n 2,
£
3, 4, 6 or 8.
As for quadrangles, the question of the existence of thick generalised n-gons
(for n 3) with s ¬nite and t in¬nite is completely open. Of course, n must be
i
even in such a geometry!

Exercises
1. Prove the assertions claimed to be “easy” in the text.
2. Construct in¬nite “free” generalised n-gons for any n 3.
i


8.9 Some generalised hexagons
In this section, we use triality to construct a generalised hexagon called G2 F 

over any ¬eld F. The construction is due to Tits. The name arises from the fact
that the automorphism groups of these hexagons are the Chevalley groups of type
G2 , as constructed by Chevalley from the simple Lie algebra G2 over the complex
numbers.
We begin with the triality quadric . Let v be a non-singular vector. Then
I
vX is a rank 3 quadric. Its maximal t.s. subspaces are planes, and each lies in
cH
I
a unique solid of each family on . Conversely, a solid on meets vX in a plane.
I I
Thus, ¬xing v, there is are bijections between the two families of solids and the
set of planes on vX . On this set, we have the structure of the dual polar
aI eI
H
£
space induced by the quadric ; in other words, the POINTs are the planes on
aI
this quadric, the LINES are the lines, and incidence is reversed inclusion. Call
this geometry . E
Applying triality, we obtain a representation of using all the points and
E
some of the lines of . I
Now we take a non-singular vector, which may as well be the same as the
vector v already used. (Since we have applied triality, there is no connection.)
134 8. The Klein quadric and triality

The geometry consists of those points and lines of which lie in vX . Thus, it
“ E
consists of all the points, and some of the lines, of the quadric . ”I
a

Theorem 8.14 is a generalised hexagon.


Proof First we observe some properties of the geometry , whose points and E
lines correspond to planes and lines on the quadric . The distance between two •I
a
points is equal to the codimension of their intersection. If two planes of meet aI
non-trivially, then the corresponding solids of (in the same family) meet in a I
line, and so (applying triality) the points are perpendicular. Hence:
(a) Points of lie at distance 1 or 2 if and only if they are perpendicular.
E

Let x y z w be four points of forming a 4-cycle. These points are pairwise E
¤¤¤
perpendicular (by (a)), and so they span a t.s. solid S. We prove:
(b) The geometry induced on S by is a symplectic GQ.
E

Keep in mind the following transformations:
solid S
point p (by triality)
h
quadric ¯ in pX p (residue of p)
h I &
PG 3 F (Klein correspondence).
h 
¤

Now points of S become solids of one family containing p, then planes of one
family in ¯ , then points in PG 3 F ; so we can identify the two ends of this chain.
I 
¤
Lines of in S become lines through p perpendicular to v, then points of ¯

E I
¯
perpendicular to v v pQ p, then t.i. lines of a symplectic GQ, by the corre-
QP P &
¤
£
spondence described in Section 8.3. Thus (b) is proved.
A property of established in Proposition 7.9 is:
E

(c) If x is a point and L a line, then there is a unique point of L nearest to x.
We now turn our attention to , and observe ¬rst:


(d) Distances in are the same as in .
“ E

For clearly distances in are at least as great as those in , and two points of
“ E “
at distance 1 (i.e., collinear) in are collinear in . E “
W
Suppose that x y lie at distance 2 in . They are joined by more than
“ E
¤
one path of length 2 there, hence lie in a solid S carrying a symplectic GQ, as
8.9. Some generalised hexagons 135

in (b). The points of in S are those of S vX , a plane on which the induced
“ H
substructure is a plane pencil of lines of . Hence x and y lie at distance 2 in . “ “
W
Finally, let x y lie at distance 3 in . Take a line L of through y; there
“ E “
¤
is a point z of (and hence of ) on L at distance 2 from x (by (c)). So x and y
E “
lie at distance 3 in . “
In particular, property (c) holds also in . “

(e) For any point x of , the lines of through x form a plane pencil.
“ “
For, by (a), the union of these lines lies in a t.s. subspace, hence they are coplanar;
there are no triangles (by (c)), so this plane contains two points at distance 2; now
the argument for (d) applies.
Finally:
(f) is a generalised hexagon.

We know it has diameter 3, and (GP2) is clearly true. A circuit of length less than
6 would be contained in a t.s. subspace, leading to a contradiction as in (d) and
(e). (In fact, by (c), it is enough to exclude quadrangles.)
Cameron and Kantor [12] give a more elementary construction of this hexagon.
Their construction, while producing the embedding in , depends only on prop- ”I
a
erties of the group PSL 3 F . However, the proof that it works uses both counting

¤

arguments and arguments about ¬nite groups; it is not obvious that it works in
general, although the result remains true.
If F is a perfect ¬eld of characteristic 2 then, by Theorem 8.5, is isomorphic –I
a
to the symplectic polar space of rank 3; so is embedded as all the points and “
some of the lines of PG 5 F . 
¤

Two further results will be mentioned without proof. First, if the ¬eld F has an
automorphism of order 3, then the construction of can be “twisted”, much as “
can be done to the Klein correspondence to obtain the duality between orthogonal
and unitary quadrangles (mentioned in Section 8.3), to produce another gener-
alised hexagon, called 3 4 F . In the ¬nite case, 3 4 q3 has parameters s q3 ,
D D  
  £
t q.
£
Second, there is a construction similar to that of Section 8.4. The generalised
hexagon G2 F is self-dual if F is a perfect ¬eld of characteristic 3, and is self-

polar if F has an automorphism σ satisfying σ2 3. In this case, the set of absolute

£
points of the polarity is an ovoid, a set of pairwise non-collinear points meeting
every line of , and the group of collineations commuting with the polarity has

as a normal subgroup the Ree group 2 2 F , acting 2-transitively on the points of
G 

the ovoid.
136 8. The Klein quadric and triality

Exercise
1. Show that the hexagon has two disjoint planes E and F, each of which

consists of pairwise non-collinear (but perpendicular) points. Show that each point
of E is collinear (in ) to the points of a line of F, and dually, so that E and F

are naturally dual. Show that the points of E F, and the lines of joining
‚ “
their points, form a non-thick generalised hexagon which is the ¬‚ag geometry of
PG 2 F . (This is the starting point in the construction of Cameron and Kantor

¤

referred to in the text.)
9

The geometry of the Mathieu groups

The topic of this chapter is something of a diversion, but is included for two rea-
sons: ¬rst, its intrinsic interest; and second, because the geometries described here
satisfy axioms not too different from those we have seen for projective, af¬ne and
polar spaces, and so they indicate the natural boundaries of the theory.


9.1 The Golay code
The basic concepts of coding theory were introduced in Section 3.2, where
we also saw that a non-trivial perfect 3-error-correcting code must have length 23
(see Exercise 3.2.2). Such a code C may be assumed to contain the zero word (by
translation), and so any other word has weight at least 7; and
223
  ¡¢  ¡
212 §
C £ £ £ £
23 23 23 23
0 1 2 3
¦¤
¥ ¦¤
¥ ¦¤
¥ ¤
We extend C to a code C of length 24 by adding an overall parity check; that
is, we put a 0 in the 24th coordinate of a word whose weight (in C) is even, and a 1
in a word whose weight is odd. The resulting code has all words of even weight,
and hence all distances between words even; since adding a coordinate cannot
decrease the distance between words, the resulting code has minimum distance 8.
In this section, we outline a proof of the following result.
Theorem 9.1 There is a unique code with length 24, minimum distance 8, and
containing 212 codewords one of which is zero (up to coordinate permutations).
This code is known as the (extended binary) Golay code. It is a linear code
(the linearity does not have to be assumed).

137
138 9. The geometry of the Mathieu groups

Remark There are many constructions of this code; for an account of some of
these, see Cameron and Van Lint [F]. As a general principle, a good construction
of an object leads to a proof of its uniqueness (by showing that it must be con-
structed this way), thence to a calculation of its automorphism group (since the
object is uniquely built around a starting con¬guration, and so any isomorphism
between such starting con¬gurations extends uniquely to an automorphism), and
gives on the way a subgroup of the automorphism group (consisting of the auto-
morphism group of the starting con¬guration). This point will not be laboured
below, but the interested reader may like to examine this and other constructions
from this point of view. The particular construction given here has been chosen
for two reasons: ¬rst, as an application of the Klein correspondence; and second,
since it makes certain properties of the automorphism group more accessible.


Proof First, we review the isomorphism between PSL¨ 4© 2 and A8 outlined in
Exercise 8.1.1. Let U be the binary vector space consisting of words of even
weight and length 8, Z the subspace consisting of the all-zero and all-one words,
¡
and V U Z. The function mapping a word of U to 0 or 1 according as its

weight is congruent to 0 or 2 mod 4 induces a quadratic form f on V , whose zeros
form the Klein quadric ; let W be the vector space of rank 4 whose lines are

bijective with the points of . Note that the points of correspond to partitions
 

 §
§§
of N 1© 8 into two subsets of size 4.
©
¡
Let „¦ N W . This set will index the coordinates of the code C we construct.

A words of C will be speci¬ed by its support, a subset of N and a subset of W . In
/
particular, 0© N W and N W will be words; so we can complement the subset of
© 
N or the subset of W de¬ning a word and obtain another word.
The ¬rst non-trivial class of words is obtained by combining the empty subset
of N (or the whole of N) with any hyperplane in W (or its coset).
A complementary pair of 4-subsets of N corresponds to a point of , and
hence to a line L in W . Each 4-subset of N, together with any coset of the corre-
sponding L, is a codeword. Further words are obtained by replacing the coset of
L by its symmetric difference with a coset of a hyperplane not containing L (such
a coset meets L in two vectors).
A 2-subset of N, or the complementary 6-subset, represents a non-singular
point, which translates into a symplectic form b on W . The quadric associated
with any quadratic form which polarises to b, together with the 2-subset of N,
de¬nes a codeword.
9.2. The Witt system 139

This gives us a total of
 
8 8 ¡
212
4 4 15 4 4 7 16 4
 ¨!   
4 2
¥ ¥ ¥ ¥

codewords. Moreover, a fairly small amount of case checking shows that the code
is linear. Its minimum weight is visibly 8.
We now outline the proof that there is a unique code C of length 24, cardinality
12 , and minimum weight 8, containing 0. Counting arguments show that such a
2
code contains 759 words of weight 8, 2576 of weight 12, 759 of weight 16, and
the all-1 word 1 of weight 24. Now, if the code is translated by any codeword,
the hypotheses still hold, and so the conclusion about weights does too. Thus,
the distances between pairs of codewords are 0, 8, 12, 16, and 24. It follows that
all inner products are zero, so C C# ; it then follows from the cardinality that
"
¡
C C# , and in particular C is a linear code.
Let N be an octad, and W its complement. Restriction of codewords to N
gives a homomorphism θ from C to a code of length 8 in which all words have
even weight. It is readily checked that every word of even weight actually occurs.
So the kernel of θ has rank 5. This kernel is a code of length 16 and minimum
weight 8. There is a unique code with these properties: it consists of the all-zero
and all-one words, together with the characteristic functions of hyperplanes of a
rank 4 vector space. (This is the ¬rst-order Reed“Muller code of length 16.) Thus
we have identi¬ed W with a vector space, and found the ¬rst non-trivial class of
words in the earlier construction.
Now, to be brief: if B is an octad meeting N in four points, then B W is a line; $
  ¡% 
if B N 2, then B W is a quadric; and all the other details can be checked,
$ $
given suf¬cient perseverence.
The automorphism group of the extended Golay code is the 54-transitive Math-
ieu group M24 . This is one of only two ¬nite 5-transitive groups other than sym-
metric and alternating groups; it is one of the ¬rst of the 26 “sporadic” simple
groups to be found; and its geometry is the starting point for the construction of
many other sporadic groups (the Conway and Fischer groups and the “Monster”).
The group M24 will be considered further in Section 9.4.


9.2 The Witt system
Let X be the set of coordinate positions of the Golay code G. Now any word
can be identi¬ed uniquely with the subset of X consisting of the positions where
140 9. The geometry of the Mathieu groups

it has entries equal to 1 (its support). Let be the set of supports of the 759
&
codewords of weight 8. An element of is called an octad; the support of a word
&
of weight 12 in G is called a dodecad.
From the linearity of G, we see that the symmetric difference of two octads is
the support of a word of G, necessarily an octad, a dodecad, or the complement of
an octad; the intersection of the two octads has cardinality 4, 2 or 0 respectively.
Three pairwise disjoint octads form a trio. (In our construction of the extended
Golay code in the last section, the three “blocks” of eight coordinates form a trio.)

Proposition 9.2 X is a 5-¨ 24© 8© 1 design or Steiner system.
¨ ('©
&

Proof As we have just seen, it is impossible for two octads to have more than
four points in common, so ¬ve points lie in at most one octad. Since there are 759£ £ ¡
octads, the average number containing ¬ve points is 759 8 24 1; so ¬ve
 
5¤ 5 ¤
points lie in exactly one octad. However, the proposition follows more directly
from the properties of the code G.
Take any ¬ve coordinates, and delete one of them. The remaining coordinates
support a word v of weight 4. But the Golay code obtained by deleting a coor-
dinate from G is perfect 3-error-correcting, and so contains a unique word c at
distance 3 or less from v. It must hold that c has weight 7 and its support contains
that of v (and c is the unique such word). Re-introducing the deleted coordinate
(which acts as a parity check for the Golay code), we obtain a unique octad con-
taining the given 5-set.

This design is known as the Witt system; Witt constructed it from its automor-
phism group, the Mathieu group M24 , though nowdays the procedure is normally
reversed. ¡
Now choose any three coordinates, and call them ∞1 , ∞2 , ∞3 . Let X )

X ∞1 ∞2 ∞3 , and let be the set of octads containing the chosen points,
)&
0 © © 
with these points removed. Then X is a 2-(21, 5, 1) design, that is, a pro-
 ) '© )
¨ &
jective plane of order 4. Since there is a unique projective plane of order 4 (see
Exercise 4.3.6), it is isomorphic to PG¨ 2© 4 .

Proposition 9.3 The geometry whose varieties are all subsets of X of cardinalities
1, 2, 3 and 4, and all octads, with incidence de¬ned by inclusion, belongs to the
diagram 1 1 1 1 1
c c c §
9.2. The Witt system 141

The remaining octads can be identi¬ed with geometric con¬gurations in PG¨ 2© 4 .
We outline this, omitting detailed veri¬cation. In fact, the procedure can be re-
versed, and the Witt system constructed from objects in PG¨ 2© 4 . See L¨ neburg [N]
u
for the details of this construction.
1. An octad containing two of the three points ∞i corresponds to a set of six
points of PG¨ 2© 4 meeting any line in 0 or 2 points, in other words, a hyperoval.
All 168 hyperovals occur in this way. If we call two hyperovals “equivalent” if
their intersection has even cardinality, we obtain a partition into three classes of
size 56, corresponding to the three possible pairs of points ∞i ; so this partition can
be de¬ned internally.
2. An octad containing one point ∞i corresponds to a set of seven points
of PG¨ 2© 4 meeting every line in 1 or 3 points, that is, a Baer subplane (when
equipped with the lines meeting it in three points). Again, all 360 Baer subplanes
occur, and the partition can be intrinsically de¬ned.
3. An octad containing none of the points ∞i is a set of eight points of PG¨ 2© 4
which is the symmetric difference of two lines. Every symmetric difference of two
lines occurs (there are 210 such sets).
Since octads and dodecads also intersect evenly, we can extend this analysis
to dodecads. Consider a dodecad containing ∞1 , ∞2 and ∞3 . It contains nine
points of PG¨ 2© 4 , meeting every line in 1 or 3 points. These nine points form
a unital, the set of absolute points of a unitary polarity (or the set of zeros of a
non-degenerate Hermitian form). Their intersections of size 3 with lines form a
2-¨ 9© 3© 1 design, a Steiner triple system which is isomorphic to AG¨ 2© 3 , and
is also famous as the Hessian con¬guration of in¬‚ection points of a non-singular
cubic. (Since the ¬eld automorphism of GF¨ 4 is ± ±2 , the Hermitian form
23
x0 x± x1 x± x2 x± is a cubic form, and its zeros form a cubic curve; in this special
1 0 2
¥ ¥
case, every point is an in¬‚ection.)


Exercises

1. Verify the connections between octads and dodecads and con¬gurations in
PG¨ 2© 4 claimed in the text.
¡
2. Let B be an octad, and Y X B. Consider the geometry whose points
0 4
are those of Y ; whose lines are all pairs of points; whose planes are all sets B) B,
0
where B) is an octad meeting B in four points; and whose solids are the octads
disjoint from B. prove that is the af¬ne geometry AG¨ 4© 2 .
4
142 9. The geometry of the Mathieu groups

9.3 Sextets
A tetrad is a set of four points of the Witt system. Any tetrad is contained
in ¬ve octads, which partition the remaining twenty points into ¬ve tetrads. Now
the symmetric difference of two octads intersecting in a tetrad is an octad; so the
union of any two of our six tetrads is an octad. A set of six pairwise disjoint tetrads
with this property is called a idxsextet.

Proposition 9.4 Let be the geometry whose POINTS, LINES and PLANES are
4
the octads, trios and sextets respectively, with incidence de¬ned as follows: a
LINE is incident with any POINT it contains; a PLANE is incident with a POINT
which is the union of two of its tetrads; and a PLANE is incident with a LINE if it
is incident with each POINT of the LINE. Then belongs to the diagram
4
1 1 1
L
©
1 1
L
where is the linear space consisting of points and lines of PG¨ 3© 2 .

Proof Calculate residues. Take ¬rst a PLANE or sextet. It contains six tetrads;
the union of any two of them is a POINT, and any partition into three sets of two
¡ ¡
is a LINE. This is a representation of the unique GQ with s t 2 that we saw in
Section 7.1.
Now consider the residue of a POINT or octad. We saw in Exercise 9.2.2
that the complement of an octad carries an af¬ne space AG¨ 4© 2 ; LINEs incident
with the POINT correspond to parallel classes of planes in the af¬ne space, and
PLANEs incident with it to parallel classes of LINEs. Projectivising and dualis-
ing, we see the points and lines of PG¨ 3© 2 .
Finally, any POINT and PLANE incident with a common LINE are incident
with one another.

The geometry does not contain objects which would correspond to the planes
of PG¨ 3© 2 in the residue of a point. The diagram is sometimes drawn with a
“ghost node” corresponding to these non-existent varieties.

Exercise
1. In the geometry of Proposition 9.4, de¬ne the distance between two
4
points to be the number of lines on a shortest path joining them. Prove that, if x is
a point and L a line, then there is a unique point of L at minimum distance from x.
9.4. The large Mathieu groups 143

9.4 The large Mathieu groups
Just as every good construction of the Golay code or the Witt system contains
the seeds of a uniqueness proof (as we observed in Section 9.1), so every good
uniqueness proof contains the seeds of an argument establishing various properties
of its automorphism group (in particular, its order, and some large subgroup, the
particular subgroup depending on the construction used). I will outline this for the
construction of Section 9.1.

Theorem 9.5 The automorphism group of the Golay code, or of the Witt system,
§
is a 5-transitive simple group of order 24 23 22 21 20 48
    


Remark This group is of course the Mathieu group M24 . Part of the reason for
the construction we gave (not the simplest available!) is that it makes our job now
easier.

Proof First note that the design and the code have the same automorphism group;
for the code is spanned by the design, and the design is the set of words of weight
8 in the code.
The uniqueness proof shows that the automorphism group is transitive on oc-
tads. For, given two copies of the Golay code, and an octad in each, there is an
isomorphism between the two codes mapping the chosen octad in the ¬rst to that
in the second. Also, the stabiliser of an octad preserves the af¬ne space structure
on its complement, and (from the construction) induces AGL¨ 4© 2 on it. (It in-
duces A8 on the octad, the kernel of this action being the translation group of the
af¬ne space.) This gives the order of the group.
Given two 5-tuples of distinct points, each lies in a unique octad. There is an
automorphism carrying the ¬rst octad to the second; then, since A8 is 5-transitive,
we can ¬x the second octad and map the 5-tuple to the correct place. The 5-
transitivity follows. ¡
We also have a subgroup H AGL¨ 4© 2 of our unknown group G, and it is
easily seen that H is maximal. Suppose that N is a non-trivial normal subgroup of
¡
G. Then HN G, and H N is a normal subgroup of H, necessarily the identity or
$
¡ ¡
the translation group. (If H N H then N G.) This gives two possibilities for
$
the order of N, namely 759 and 759 16. But N, a normal subgroup of a 5-transitive

group, is at least 4-transitive, by an old theorem of Jordan; so 24 23 22 21 divides
  
   
N , a contradiction. We conclude that G is simple.
144 9. The geometry of the Mathieu groups

The stabiliser of three points is a group of collineations of PG¨ 2© 4 , neces-
sarily PSL¨ 3© 4 (by considering order). The ovals and Baer subplanes each fall
into three orbits for PSL¨ 3© 4 , these orbits being the classes used in L¨ neburg™s
u
construction. The set-wise stabiliser of three points is P“L 3© 4 . Looked at an-
¨
other way, L¨ neburg™s construction and uniqueness proof gives us the subgroup
u
P“L¨ 3© 4 of M24 .


9.5 The small Mathieu groups
To conclude this chapter, I describe brie¬‚y the geometry associated with the
Mathieu group M12 .
There are two quite different approaches. One locates the geometry within the
Golay code. The group M12 can be de¬ned as the stabiliser of a dodecad in M24 ;
it acts sharply 5-transitively on this dodecad, and on the complementary dodecad,
but the two permutation representations are not equivalent. The dodecad D carries
a design, which can be seen as follows. It intersects any octad in an even number,
at most 6, of points; and any ¬ve points of D lie in a unique octad, meeting D
in 6 points. So the intersections of size 6 of octads with D are the blocks of a
5-¨ 12© 6© 1 design or Steiner system.
Alternatively, there are “characteristic 3” objects with properties resembling
the binary Golay code. There is a ternary Golay code, a set of ternary words of
length 12 (that is, entries in GF 3 ) forming a subspace of GF¨ 3 12 of rank 6, and
¨
having minimum weight 6; the supports of weight 6 of codewords form the blocks
of the design. Alternatively, there is a set of 12 points in PG¨ 5© 3 on which M12 is
induced, as follows. There is a Hadamard matrix H of size 12 12 (a matrix with
5
¡
entries 1 satisfying HH 12I), unique up to row and column permutations
6 7
and sign changes; over GF 3 , it has rank 6, and its rows span the required points.
¨
Now the design is obtained as follows. The point set is identi¬ed with the set of
rows. Any two columns agree in six rows and disagree in the other six, de¬ning
£ ¡
two sets of size 6 which are blocks of the design; and all 2 12 132 blocks are
 2 ¤
obtained in this way.
Some connection between characteristics 2 and 3 can be seen from the obser-
vation we made in Section 9.2, that a unital in PG¨ 2© 4 is isomorphic to the af¬ne
plane AG¨ 2© 3 . It turns out that the three times extensions of these two planes
are associated with codes in characteristics 2 and 3 respectively, and that one ex-
tension contains the other. However, the large Witt system is not embeddable in
PG¨ 5© 4 , so the analogy is not perfect.
9.5. The small Mathieu groups 145

Exercise
¡   8 
¡
1. Let G AG¨ 2© 3 , and X the set of lines of G (so that X 12). Consider
the subsets of X of the following types:
9
all unions of two parallel classes;
9
the lines of two classes containing a point p, and those of the other two not
containing p;
9
a parallel class, with the lines of the others containing a ¬xed point p; and
the complements of these.
¡
Show that these 6 54 2 36 132 sets of size 6 form a 5-¨ 12© 6© 1 design.

¥ ¥
Assuming the uniqueness of this design, prove that AGL¨ 2© 3 M12 .
A
@
10

Exterior powers and Clifford
algebras

In this chapter, various algebraic constructions (exterior products and Clifford al-
gebras) are used to embed some geometries related to projective and polar spaces
(subspace and spinor geometries) into projective spaces. In the process, we learn
more about the geometries themselves.


10.1 Tensor and exterior products
Throughout this chapter, F is a commutative ¬eld (except for a brief discussion
of why this assumption is necessary).
 
The tensor product V W of two F-vector spaces V and W is the free-est
bilinear product of V and W : that is, if (as customary), we write the product of
 
vectors v V and w W as v w, then we have
¡ ¡
¢ ¢ ¢
±v w ±v
         
v1 v2 w v1 w v2 w¦ w
¥ ¥ ¦¤
£ ¤ £ ¤
¢ ¢ ¢
v ±w ±v
         
v w1 w2 v w1 v w2 w
¥¤ ¦ ¥¤
£ £ ¨¤
§

Formally, we let X be the F-vector space with basis consisting of all the ordered
¢
pairs v¦ w (v V¦ w W ), and Y the subspace spanned by all expressions of the
¡ ¡
¤
¢ ¢ ¢  
form v1 v2 w v1 w v2 w and three similar expressions; then V W
¦ ¦ ¦ ¥
£ ©¤ ¤
© ¤
¢
 
X Y , with v w the image of v¦ w under the canonical projection. Sometimes,
 ¤
 
to emphasize the ¬eld, we write V F W .
This construction will only work as intended over a commutative ¬eld. For
¢ ¢ ¢ ¢
±β v ± βv βv ±w βv ±w β± v
         
w w w
¥¤ ¥¤ ¥ ¥¤ ¦¤

147
148 10. Exterior powers and Clifford algebras

so if v w 0 then ±β β±.
 
¥ ¥
There are two representations convenient for calculation. If V has a basis
   
v1 vn and W a basis w1 wm , then V W has a basis
¦ § ¦ ¦ § ¦
§§  §§ 
  
vi wj : 1 i n¦ 1 j m
    §
 
If V and W are identi¬ed with F n and F m respectively, then V W can be
 
identi¬ed with the space of n m matrices over F, where v w is mapped to the 
matrix v w. ¢ ¢ ¢
 
In particular, rk V W rk V rk W . ¥¤ !¤ ¤
Suppose that V and W are F-algebras (that is, have an associative multipli-
 
cation which is compatible with the vector space structure). Then V W is an
algebra, with the rule
¢ ¢ ¢ ¢
     
v1 w1 v2 w2 v1 v2 w1 w2
¥¤
!¤ ¤ ¨¤
§

Of course, we can form the tensor product of a space with itself; and we can
form iterated tensor products of more than two spaces. Let k V denote the k-fold "
tensor power of V . Now the tensor algebra of V is de¬ned to be

#
¢ ¢ k
TV V
¥¤ " ¦¤
k$ 0

with multiplication given by the rule
¢ ¢
           
v1 vn vn% vm% v1 vm%
1 n n
¥¤
§
§§ !¤ §
§§ §
§§

on homogeneous elements, and extended linearly. It is the free-est associative
algebra generated by V .
The exterior square of a vector space V is the free-est bilinear square of V in
which the square of any element of V is zero. In other words, it is the quotient of
2
V by the subspace generated by all vectors v v for v V . We write it as 2 V ,
 
" ¡ &
or V V , and denote the product of v and w by v w. Note that w v v w. If
' ' ' ¥ '
©

v1 vn is a basis for V , then a basis for V V consists of all vectors v1 v j ,
¦ § ¦ ' '
§§ 
for 1 i j n; so
( 

n
¢ ¢
1
rk V V 2n n 1
' 0¥ ¤
) ¥ © ¨¤
§
21

More generally, we can de¬ne the kth exterior power k V as a k-fold multi- &
linear product, in which any product of vectors vanishes if two factors are equal.
10.1. Tensor and exterior products 149

Its basis consists of all expressions vi1 vik , with 1 i1 ik n; and

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