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Finite Geometry, Dirac Groups and the Table of Real Clifford Algebras

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Abstract

Associated with the real Clifford algebra Cl(p, q), p + q = n, is the finite Dirac group G(p, q) of order 2n+1. The quotient group V n = G(p, q)/ *#x007B;± 1*#x007D;, viewed additively, is an ndimensional vector space over GF(2) = *#x007B;0, 1*#x007D; which comes equipped with a quadratic form Q and associated alternating bilinear form B. Properties of the finite geometry over GF(2) of V nB, Q — in part familiar, in part less so — are given a rather full description, and a dictionary of translation into their Dirac group counterparts is provided. The knowledge gained is used, in conjunction with facts concerning representations of G(p, q), to give a pleasantly clean derivation of the well-known table of Porteous (1969/1981) of the algebras Cl(p, q). In particular the finite geometry highlights the “antisymmetry” of the table about the column p - q = -1. Several low-dimensional illustrations are given of the application of finite geometry results to Dirac groups. Particular emphasis is laid on certain interesting finite geometry symmetry methods, the latter being given a rather full treatment in the appendices. Finite geometry is also used to study the automorphisms of the Dirac groups, and the splitting of certain exact sequences.
FINITE GEOMETRY, DIRAC GROUPS AND
THE TABLE OF REAL CLIFFORD
ALGEBRAS
R. SHAW
Abstract
Associated with the real Clifford algebra Cl(p, q), p+q = n, is the finite
Dirac group G(p, q) of order 2
n+1
. The quotient group V
n
= G(p, q)/1},
viewed additively, is an n-dimensional vector space over GF(2) = {0, 1}
which comes equipped with a quadratic form Q and associated alternating
bilinear form B. Properties of the finite geometry over GF(2) of V
n
, B, Q
in part familiar, in part less so are given a rather full description,
and a dictionary of translation into their Dirac group counterparts is pro-
vided. The knowledge gained is used, in conjunction with facts concerning
representations of G(p, q), to give a pleasantly clean derivation of the well-
known table of Porteous (1969/1981) of the algebras Cl(p, q). In particular
the finite geometry highlights the “antisymmetry” of the table about the
column p q = 1. Several low-dimensional illustrations are given of the
application of finite geometry results to Dirac groups. Particular empha-
sis is laid on certain interesting finite geometry symmetry methods, the
latter being given a rather full treatment in the appendices. Finite geom-
etry is also used to study the automorphisms of the Dirac groups, and the
splitting of certain exact sequences.
inite geometry, Clifford algebras, Dirac groups, Caps on quadrics, Con-
well heptads, Non-polar sets, Symmetric groups, Split exact sequences
1 Introduction
1.1 The GF(2) Connection
Let Cl(p, q) denote the universal Clifford algebra for a real n-dimensional or-
thogonal space IR
p,q
of signature (p, q). If {e
1
, e
2
, . . . , e
n
} is a fixed choice of or-
thonormal basis for IR
p,q
Cl(p, q), then the e
i
(together with 1, if n = p = 1)
generate, under Clifford multiplication, a finite group G(p, q) called the Dirac
group of Cl(p, q), whose order is 2
n+1
:
G(p, q) = 1, ±e
i
, ±e
i
e
j
, . . . , ±w}, (i < j < . . .) (1.1)
1
where w = e
1
e
2
. . . e
n
. Observe that every commutator ghg
1
h
1
and every
square g
2
in G(p, q) is equal to +1 or 1. Consequently the quotient group
G(p, q)/1} = V
n
(1.2)
by the central subgroup {1, 1} is an elementary abelian 2-group V
n
of order
2
n
. We will therefore view V
n
additively, as a vector space of dimension n over
the finite field GF(2) = IF
2
= {0, 1}. If for the central extension
1 1} G(p, q) V
n
1
we choose a section e: V
n
G(p, q): x 7→ e
x
, we can display the Dirac group in
the manner
G(p, q) = e
x
: x V
n
}. (1.3)
Then e
x
e
y
= (1)
f(x,y)
e
x+y
, for some function f : V
n
× V
n
GF(2) which
depends on our choice of section. However in
(e
x
)
2
= (1)
Q(x)
, x V
n
, (1.4)
the function Q : V
n
GF(2) is independent of our choice of section. Since
(e
x
e
y
)
2
= (e
x+y
)
2
, it follows from (1.4) that we have the commutator result
e
x
e
y
(e
x
)
1
(e
y
)
1
= (1)
B(x,y)
(1.5)
where B : V
n
× V
n
GF(2) is given by B(x, y) = Q(x + y) + Q(x) + Q(y) and
is thus also independent of our choice of section. Clearly B(x, x) = 0; in fact
an easy check shows that B is a bilinear form on V
n
. Consequently, as after eq.
(2.1) below, Q is a quadratic form on V
n
having B as its associated alternating
bilinear form.
Is B non-degenerate? The answer depends on whether n is even or odd. If
n is even then G(p, q) has centre 1}, which fact entails that the only vector
k V
n
which satisfies B(k, x) = 0 for all x V
n
is the zero vector, i.e., B is
non-degenerate. If n = 2m + 1 is odd then B, being alternating, is necessarily
degenerate. However it is still of maximal rank, namely 2m. For G(p, q) now
has centre 1, ±w}, which fact entails, on writing ±w = ±e
k
, that B has
1-dimensional kernel spanned by k.
Remark Clifford algebraists and finite geometers frequently carry out the
same computations without realizing it the former proceeding multiplica-
tively using commutativity/anticommutativity and the latter proceeding ad-
ditively using orthogonality/non-orthogonality. As an elementary example, in
Clifford algebra one argues that if e
x
anticommutes with both e
y
and e
z
, then
e
x
commutes with their product e
y
e
z
; in finite geometry (over GF(2)) one ar-
gues that if x is non-orthogonal to both y and z, then x is orthogonal to their
sum y + z. (For from B(x, y) = 1 and B(x, z) = 1 it follows by bilinearity that
B(x, y + z) = 1 + 1 = 0.) Similarly all Clifford algebraists will (time and time
2
again!) have computed the square of the volume element w = e
1
e
2
. . . e
n
and
arrived at the result
w
2
= (1)
q+m
, where m = [n/2]. (1.6)
For the finite geometry computation, see the proof, below, of lemma 4.6.
As a third example, recall the notions of Eddington (1936), in the case n = 4,
of pentads, triads, conjugate triads and anti-triads. In finite geometry terms a
pentad is what in section 3 we term a MOS for a space V
4
which carries Sp(4, 2)-
geometry. A special feature of dimension n = 4 is that the MOS’s coincide with
the elliptic quadrics supported by the given Sp(4, 2)-geometry, and so Edding-
ton’s six pentads are essentially the six elliptic quadrics which are so supported,
see lemma 2.1. In the language of the projective space PG(3, 2) associated with
V
4
, Eddington’s triads correspond to hyperbolic lines, which number 20, viz. 10
conjugate or polar pairs {λ, λ
}, and his anti-triads correspond to the isotropic
or self-polar (λ = λ
) lines, which number 15. (See section 7 and appendix C for
details.) In the case of G(3, 1), the resulting quadric in PG(3, 2) is of hyperbolic
type and has a unique pair {λ, λ
} of external lines, and this fact goes along
with G(3, 1) possessing a unique pair of mutually centralizing Q
8
subgroups.
As a final example, consider the recent finite geometry paper of Dye (1992).
But for our above introduction, its title “Maximal sets of non-polar points of
quadrics and symplectic polarities over GF(2)” might at first sight appear un-
promising to a Clifford algebraist. In fact anyone familiar with the table (theo-
rem 5.2) of real Clifford algebras, and with the result (1.6), could have derived
finite geometry results, as in appendix A, which subsume the chief results in
Dye (1992) — provided that the relevant dictionary, see sections 5 and 6 below,
was to hand. (Nevertheless it does appear to be logically purer to proceed in
the direction finite geometry Clifford algebra, as will be done in the rest of
this paper.)
1.2 Plan
Since Clifford algebra practitioners are not necessarily knowledgeable concerning
finite geometry, it seems sensible to devote the initial sections 2, 3, 4 to a fairly
detailed description of the relevant geometry. This is all the more called for
because the geometry in question is of a rather peculiar kind, due to the ground
field being the field GF(2) comprising only the two elements {0, 1}. Moreover the
results in sections 3 and 4, and in particular theorem 4.5, which are crucial to
our Clifford algebra concerns, appear to be little known even by finite geometry
specialists. Some further relevant finite geometry results are presented in the
appendices.
In section 5 we first of all recall some well-known facts concerning the repre-
sentation theory of the finite group G(p, q). We then observe that our theorem
4.5 has already told us the Frobenius-Schur type of the (relevant) irreducible
representations of G(p, q), and note that this knowledge in turn tells us the na-
ture of the real algebra Cl(p, q), as given in theorem 5.2. Thus we see that the
3
finite geometry result in theorem 4.5 translates into the well-known table of the
real Clifford algebras (discovered by Cartan in 1908), as given by Porteous (1981,
table 13.26, p. 250). Next, in section 6, we present a dictionary of translation
between the finite geometry surrounding V
n
, B, Q and corresponding aspects
of the Dirac group G(p, q). Further applications of finite geometry to Clifford
algebras and Dirac groups are treated in sections 7 and 8, with illustrations,
chiefly in the cases of dimension n 6. Of help in the cases n = 4, 5, 6 is the
rather pretty mathematics, see appendices B — D, surrounding the finite group
isomorphisms
Sp(4, 2)
=
O(5, 2)
=
S
6
and O
+
(6, 2)
=
S
8
. (1.7)
Finally, in section 9, we make further use of finite geometry in the study of the
automorphisms of Dirac groups.
Remark Of course the fact that the Dirac group G(p, q), although finite, nev-
ertheless captures much of importance concerning the Clifford algebra Cl(p, q)
has been appreciated previously by many authors, see e.g., Eckmann (1942)
and Braden (1985). That the study of G(p, q) profitably involves finite ge-
ometry over GF(2) is also well-known: accounts of relevant mathematics can
be found dotted about the literature, cf. Quillen (1971), Griess (1973), and
Frenkel, Lepowsky & Meurman (1988). However the finite geometry derivation
in section 5 of the table of real Clifford algebras has not, it seems, appeared
previously in the literature. Moreover the author believes that the present el-
ementary, detailed and self-contained account has certain virtues of coherence
and accessibility, which may help to increase the general understanding of these
matters. Hopefully the collection in one place, in sections 2, 3, 4 and in the
appendices, of so much “GF(2) geometry” may attract other researchers to seek
out further applications of finite geometry to Clifford algebras.
2 Finite Geometry over GF(2)
Let V = V
n
= V (n, 2) denote any n-dimensional vector space over GF(2).
Due to the peculiar nature of GF(2) a one-dimensional subspace x V
is spanned by a unique nonzero vector x V. Consequently the points of the
projective geometry PG(n 1, 2) = IPV associated with V can be identified
with the nonzero vectors of V. Of course, relative to a choice of basis for V,
we have V
n
=
(IF
2
)
n
, and so |V
n
| = 2
n
and |PG(n 1, 2)| = 2
n
1. Recall
that (r + 1)-dimensional vector subspaces of V become r-dimensional projective
subspaces of IPV. Projective subspaces of dimensions r = 1, 2, . . . , n 2 are
called lines, planes, solids, . . . , hyperplanes, respectively. For example, the
projective geometry PG(4, 2) associated with V (5, 2), has 31 points, 155 lines,
155 planes and 31 solids. Due to Grassmann’s relations for the vector subspaces
of V (5, 2), a line of PG(4, 2) always meets a solid, but may or may not be skew
to a plane, of PG(4, 2).
4
2.1 Symplectic Geometry
Suppose now that V
n
is equipped with a preferred alternating bilinear form B :
V × V GF(2). Note that the alternating condition B(x, x) = 0 for all x V,
in conjunction with bilinearity, implies (since 1 = +1) that B is symmetric.
At times it proves convenient to abbreviate B(x, y) to x.y . Two vectors x, y
are said to be orthogonal, or perpendicular, whenever x.y = 0. However it seems
clear enough that over GF(2) non-orthogonality of vectors should be considered
an equally important property. For the “negative” information x.y 6= 0 implies,
over GF(2), the “positive” fact that x.y = 1. This insight is followed up in
sections 3, 4 below.
Since B is alternating its rank is even. If V has even dimension n = 2m
and if B is non-degenerate, i.e., of rank 2m, then the invariance group Sp(V )
of B is (isomorphic to) the symplectic group Sp(2m, 2). It is well-known, see
Artin (1957), that two alternating forms on a vector space V (over any field) lie
on the same GL(V )-orbit if and only if they have the same rank. In particular
GL(2m, 2) possesses a single conjugacy class of Sp(2m, 2) subgroups.
In general, given (V, B) and any subset Y V, we set Y
= {x V : x.y = 0
for all y Y }, and note that Y
is a subspace of V. The radical of V, also
termed the kernel of B, is defined to be the subspace: rad V = ker B = V
.
In applications to Clifford algebras we shall be concerned solely with the cases
(V
n
, B) for which B has maximal rank namely either the even-dimensional
case n = 2m, with ker B = {0}, or the odd-dimensional case n = 2m + 1
where B, of rank 2m, has 1-dimensional kernel = k , spanned by the unique
nonzero vector k which satisfies k.x = 0 for all x V.
2.2 Orthogonal Geometry
Suppose now that V = V (n, 2) is equipped with a preferred quadratic form
Q : V GF(2). Then Q determines an associated alternating bilinear form B
via the relation
Q(x + y) + Q(x) + Q(y) = B(x, y). (2.1)
In fact, over GF(2), a quadratic form Q on V is nothing more than any solution
Q of (2.1) for some choice of alternating bilinear form B. (One does not need
to demand Q(λx) = λ
2
Q(x) since this demand amounts merely to Q(0) = 0,
which last follows from (2.1).) Note that the linear forms f, i.e., elements of
the dual space V
, are precisely those quadratic forms whose associated bilinear
form is the zero form. (This fact may be made more palatable by remembering
that, over GF(2), f(x) = f (x)
2
.)
Now it is easy to see that, for any given alternating bilinear form B, equation
(2.1) always possesses a solution Q. Given (V, B), let Quad(B) denote the set
of all those quadratic forms Q satisfying (2.1). Then observe that
Quad(B) = {Q
0
+ f : f V
}, (2.2)
where Q
0
denotes any particular solution of (2.1), i.e. Quad(B) is a coset of V
5
in the space Quad(V ) of all quadratic forms on V. Thus the given alternating
form B on V
n
supports a family of 2
n
quadratic forms.
Consider now the even-dimensional case in which V
n
= V
2m
is equipped with
Sp(2m, 2) geometry via a non-degenerate alternating form B. Since any linear
form f can then be written
f(x) = B(a, x), for some unique a V, (2.3)
it follows that
Quad(B) = {Q
a
: a V } (2.4)
where
Q
a
(x) = Q
0
(x) + B(a, x) = Q
0
(a + x) + Q
0
(a). (2.5)
Let the equations Q
0
(x) = 0, Q
0
(x) = 1 have respectively N
+
, N
solutions
x V. Then N
+
+ N
= |V | = 2
2m
, and so
N
+
= 2
2m1
+ c, N
= 2
2m1
c,
for some integer c. Consequently Quad(B) is the disjoint union
Quad(B) = Quad
+
(B) Quad
(B) (2.6)
where
Quad
+
(B) = {Q
a
: Q
0
(a) = 0}, Quad
(B) = {Q
a
: Q
0
(a) = 1} (2.7)
and where Quad
+
(B), Quad
(B) have respective sizes
|Quad
+
(B)| = N
+
, |Quad
(B)| = N
.
Lemma 2.1 A given non-degenerate alternating form B on V
2m
supports
precisely two distinct Sp(2m, 2)-orbits Quad
(B), = ±, of quadratic forms,
with Quad
(B) consisting of N
= 2
2m1
+ 2
m1
quadratic forms Q
, each
having N
zeros.
Proof Count in two ways the number N of pairs (Q, v), Q Quad(B), v V,
such that Q(v) = 0. On the one hand we have immediately N = (N
+
)
2
+
(N
)
2
= 2
4m1
+ 2c
2
. On the other hand for fixed v V the number of
solutions Q
a
Quad(B) of Q
a
(v) = 0 equals, by (2.5), the number of solutions
a V of B(a, v) = Q
0
(v), namely 2
2m
if v = 0 and 2
2m1
if v 6= 0, and so
N = 2
2m
+ (2
2m
1)2
2m1
= 2
4m1
+ 2
2m1
. Hence c = ±2
m1
(and without
loss of generality we may assume that our original choice of Q
0
was such that
c = +2
m1
).
Finally, to show that each of Quad
(B) forms a single Sp(2m, 2)-orbit, ob-
serve from (2.1), (2.5) that if Q
0
(a) = 0 then
Q
a
(x) = Q
0
(T
a
x) (2.8)
6
where T
a
: V V is the linear mapping, termed a symplectic transvection,
defined by
T
a
x = x + B(a, x)a, x V. (2.9)
(Since both Q
0
and Q
a
satisfy (2.1), it follows from (2.8) that T
a
does indeed
preserve B.) []
Remark Consequently Sp(2m, 2) possesses two distinct conjugacy classes of
orthogonal subgroups, the index in Sp(2m, 2) of a subgroup O
(2m, 2) which
stabilizes a quadratic form Q Quad
(B) being N
, = ±. Naturally all this
applies for each Sp(2m, 2) subgroup of GL(m, 2), with all the O
(2m, 2) sub-
groups, for given , forming a single conjugacy class within GL(2m, 2). Of course
the foregoing is all very well-known. Nevertheless the above proof seems espe-
cially simple: in particular it does not use (a) a knowledge of canonical forms,
(b) induction, or (c) coordinates.
Remark For the record it is known, see Hirschfeld & Thas (1991), that any
Q Quad
+
(B) can be expressed in the coordinate canonical form
Q(x) = x
1
x
2
+ x
3
x
4
+ . . . + x
2m1
x
2m
(2.10
+
)
and any Q Quad
(B) in the form
Q(x) = (x
1
+ x
2
+ x
1
x
2
) + x
3
x
4
+ . . . + x
2m1
x
2m
. (2.10
)
In either case the associated alternating form B takes the canonical form
B(x, y) = (x
1
y
2
+ x
2
y
1
) + . . . + (x
2m1
y
2m
+ x
2m
y
2m1
) (2.11)
corresponding to a decomposition of V into an orthogonal sum of hyperbolic
planes. If Q Quad
, we write (Q) = and refer to (Q) as the type of Q.
Quadratic forms of types +1, 1 are also termed hyperbolic, elliptic, respectively.
In projective language the nonzero vectors x satisfying Q(x) = 0 are referred
to as the points on a projective quadric in PG(2m 1, 2). This quadric (still
assuming we are dealing with the case when the associated alternating form
B is non-degenerate) is said to be of the type H
2m1
or E
2m1
according as
(Q) = +1 or 1, see Hirschfeld & Thas (1991). Of course for Q of type the
projective quadric has N
1 points on it. For example in PG(5, 2) there exist,
for given B, 64 non-degenerate quadrics. Of these 2
5
+ 2
2
= 36 are H
5
quadrics,
each having 35 points, and 2
5
2
2
= 28 are E
5
quadrics, each having 27 points.
Consider also the odd-dimensional case in which V
n
= V
2m+1
is equipped
with an alternating form B having (maximal) rank 2m and one-dimensional
kernel k . Solutions Q of (2.1) such that Q(k) = 1 are said to be of parabolic
type. In projective language, see Hirschfeld & Thas (1991), such a Q defines
a P
2m
quadric with nucleus k. Solutions Q of (2.1) such that Q(k) = 0 fall
into two kinds: in projective language they define hyperbolic and elliptic cones,
Π
0
H
2m1
and Π
0
E
2m1
, whose vertex Π
0
is k. Unlike the parabolic quadrics,
the latter are singular in that their equations can be expressed using fewer
7
than the 2m + 1 coordinates. The non-singular quadratic forms on V
2m+1
of
parabolic type give rise to a single conjugacy class of O(2m + 1, 2) subgroups in
GL(2m + 1, 2). By considering the induced non-degenerate Sp(2m, 2)-geometry
of the quotient space V
2m+1
/ k one easily derives the isomorphism
O(2m + 1, 2)
=
Sp(2m, 2).
In the odd-dimensional case under consideration we define the type (Q) of Q
to be +1, 0 or 1 according as Q is of the forgoing hyperbolic, parabolic or
elliptic types, respectively. By making use of the knowledge gained in lemma
2.1 one easily derives the next lemma.
Lemma 2.2 The three kinds (Q) = +1, 0, 1 of quadratic form Q on
V (2m+1, 2) which are supported by an alternating form B of maximal rank 2m
are distinguished by the property that Q(x) = 0 has 2
2m
+ (Q)2
m
solutions. A
given B supports 2
2m
: Q
0
s of the parabolic type = 0 and 2
2m1
+ 2
m1
: Q
0
s
of type = ±. []
Remark Canonical forms for cones of kinds Π
0
H
2m1
and Π
0
E
2m1
are
precisely as in equations (2.10), while a canonical form for a P
2m
quadric is
given by
Q(x) = x
1
x
2
+ x
3
x
4
+ . . . + x
2m1
x
2m
+ x
2m+1
. (2.12)
(The coordinates are such that k = (0, . . . , 0, 1) is the vertex of the cones and
also the nucleus of the parabolic quadric.) In the case of a quadratic form Q on
V
n
whose alternating form B is of maximal rank, let
δ(Q) = (no. of zeros of Q) (no. of non-zeros of Q). (2.13)
>From lemmas 2.1, 2.2 we see that
δ(Q) =
(Q)2
m
, if n = 2m,
(Q)2
m+1
, if n = 2m + 1.
(2.14)
The following lemma, whose proof is immediate, will also prove useful.
Lemma 2.3 Let B be of maximal rank on V
n
. For Q
0
Quad(B) and
a V
n
, let Q
a
Quad(B) be defined as in (2.5) by Q
a
(x) = Q
0
(x) + B(a, x) =
Q
0
(a + x) + Q
0
(a). Then
(Q
a
) =
+(Q
0
), if Q
0
(a) = 0,
(Q
0
), if Q
0
(a) = 1.
3 Non-orthogonality and Symplectic Geometry
Let V = V (n, 2) be equipped with a preferred alternating bilinear form B. Ini-
tially we make no restrictions on the rank of B. Consider a set S = {a
1
, a
2
, . . . , a
s
}
of vectors which are mutually non-perpendicular:
a
i
.a
j
B(a
i
, a
j
) = 1, i 6= j. (3.1)
8
We will refer to such a set as an OS (= Off-diagonal Set). This terminology may
not be very apt, for in symplectic geometry the “diagonal” numbers B(a
i
, a
i
)
are all zero for any set. Perhaps the O can also remind one that S is off-
diagonal in an Outstanding manner, and the S can also remind one that the
geometry is Symplectic. However our terminology has at any rate the virtue
that it leads to pronounceable acronyms unlike the alternatives such as PNPS
(= Pair-wise N on-Polar S et) or MNCS (= M utually N on-C onjugate Set). An
OS S in V
n
will be termed
a BOS for V
n
if S is a Basis for V
n
,
a COS if S is Complete in the sense that there does not exist x V
such that x.a
i
= 1, each i = 1, 2, . . . , s,
a DOS if S is a linearly Dependent set,
a MOS if s = |S| is Maximal for the given n = dim V.
Given the OS S = {a
1
, . . . , a
s
} V
n
, let S denote the subspace S and put
u =
P
i
a
i
S.
Then lemmas 3.1 — 3.5 below hold. Their proofs are all very easy, depending on
very little more than that a sum of t 1
0
s is, over GF(2), equal to 0 or 1 according
as t is even or odd. Theorem 3.6 is an immediate consequence of these lemmas.
Lemma 3.1 Let 0 6= x S = a
1
, . . . , a
s
. Then
(i) x.a
i
= 0, i = 1, 2, . . . , s s = 2r + 1 and x = u,
(ii) x.a
i
= 1, i = 1, 2, . . . , s s = 2r and x = u.
Proof For x = Σ
i
x
i
a
i
put σ = Σ
i
x
i
and note from (3.1) that
x.a
i
= x
i
+ σ, 1 i s. (3.2)
Now if x = u we have σ = 0 or 1 according as s is even or odd. Consequently,
for each i = 1, 2, . . . , s,
u.a
i
= 1 (s even ) and u.a
i
= 0 (s odd). (3.3)
Thus both of the implications hold. In the other direction, given that
x (6= 0) satisfies either of the conditions (i) x.a
i
= 0 (ii) x.a
i
= 1, in each case
for each i = 1, 2, . . . , s, then it follows from (3.2) that x = u; moreover, by
(3.3), s is necessarily odd in case (i) and even in case (ii). []
Lemma 3.2 Suppose that s = |S| = 2r is even. Then
(i) S is non-singular (i.e. S S
= {0}),
(ii) S is a BOS for S (and so even DOS’s do not exist),
9
(iii) S
= S {u} is the unique extension of S to a larger OS for S (and
then S
{a} is a BOS for S for any a S
),
(iv) S
is a COS for V
n
.
Proof Part (i) follows from lemma 3.1(i). Part (ii) follows from setting x = 0
in eq. (3.2). Part (iii) follows from lemma 3.1(ii). Finally, S
can not be
extended to a larger OS, since the assumptions x.a
i
= 1 entail that x.u = 0,
and so contradict the assumption x.u = 1. []
Lemma 3.3 Suppose that s = 2r + 1 (r 1) is odd and that S is a DOS.
Then
(i)
P
s
1
a
i
= 0, this being the only linear dependence amongst the a
i
S
(and so, for each i, S {a
i
} is a BOS for S and dim S = 2r),
(ii) S is non-singular,
(iii) S is a COS for V
n
.
Proof (i) We are given that x Σ
2r+1
i
x
i
a
i
= 0, for some x
i
not all zero, whence
by (3.2) the x
i
are all equal, and so
P
a
i
= 0 is the only linear dependence.
(ii) S is spanned by an even OS, e.g., S {a
1
}, whence by lemma 3.2(i) S
is non-singular.
(iii) S, being the extension of S {a
1
}, is by lemma 3.2(iv) a COS for V
n
.
[]
Lemma 3.4 Suppose that s = 2r + 1 is odd and that S is a BOS for S
(and so dim S = 2r + 1). Then
(i) rad S(= S S
) = u ,
(ii) S is a COS for S.
Proof Parts (i), (ii) follow respectively from parts (i), (ii) of lemma 3.1. []
Lemma 3.5 (i) Suppose that B is non-degenerate on V
2m
. Given an or-
thogonal decomposition V
2m
= V
2m2
V
2
let {b
1
, . . . , b
2m2
} be any BOS for
V
2m2
and {w, v} any basis for V
2
. Then {a
1
, . . . , a
2m2
, w, v} is a BOS for
V
2m
if a
i
= b
i
+ w + v.
(ii) Suppose that B has rank 2m on V
2m+1
. Given a decomposition V
2m+1
= V
2m
k , let {a
1
, . . . , a
2m
} be any BOS for V
2m
. Then {a
i
, . . . , a
2m
, k +
u} is a BOS for V
2m+1
if u =
P
i
a
i
.
Proof A straightforward check, on bearing in mind that w.v = 1. []
Theorem 3.6 If V
n
is equipped with an alternating bilinear form B of
maximal rank, then
(i) BOS’s exist for (V
n
, B),
(ii) a MOS for V
2m
has size 2m + 1 and a MOS for V
2m+1
has size 2m + 1,
(iii) if n = 2m every MOS is the extension of a BOS for V
n
,
10
(iv) if n = 2m + 1 there are two kinds of MOS:
a) the BOS’s for V
n
,
b) the MOS’s for the non-singular hyperplanes V
2m
V
n
;
moreover, for any BOS {a
1
, . . . , a
2m+1
},
P
a
i
spans rad (V
2m+1
),
(v) COS’s exist of any odd size 2r + 1, 1 r m = [n/2], but for no even
size.
Proof (i) BOS’s exist in the case of V
2
, since for any basis {w, v} we have
w.v = 1. Using lemma 3.5 the general result follows by induction on n.
(ii) In the case of V
2m
we obtain an OS of size 2m + 1 by extending any
BOS. In the case of V
2m+1
any BOS is of size 2m + 1. On the other hand OS’s
of size 2m + 2 would be even-sized DOS’s, and so ruled out by lemma 3.2(ii).
(iii) (v) now follow easily from previous results. []
Remark Rather than use induction to prove the existence of BOS’s for
(V
n
, B), one can alternatively argue as follows. In the case n = 2m let S
0
=
{a
0
1
, . . . , a
0
2m
} be any basis for V
2m
. Let B
0
be that bilinear form on V
2m
such
that B
0
(a
0
i
, a
0
i
) = 0, for each i, and B
0
(a
0
i
, a
0
j
) = 1, i 6= j. Then B
0
is seen to
be alternating and moreover, by lemma 3.2, S
0
is a BOS for (V
2m
, B
0
) and
B
0
is non-degenerate. Since B is also given to be non-degenerate, there exists
T GL(V
2m
) which sends B
0
to B. The image under T of S
0
is then a BOS
for (V
2m
, B). The case n = 2m + 1 is now quickly dealt with upon using lemma
3.5(ii).
4 BOS’s and Quadratic Forms
Let S = {a
1
, . . . , a
n
} be a BOS for (V
n
, B). Given non-negative integers p, q,
with p + q = n, there exists a unique quadratic form Q on V
n
supported by B
such that
Q(a
1
) = . . . = Q(a
p
) = 0, and Q(a
p+1
) = . . . = Q(a
n
) = 1. (4.1)
For equation (2.1) fixes Q up to a linear form, and equation (4.1) fixes the linear
form. Indeed by varying the order of the basis vectors a
i
, and varying (p, q),
we obtain in this way all the 2
n
quadratic forms Q satisfying (2.1). We will say
that Q is of type (p, q) with respect to the BOS S for (V
n
, B). We will also say
that S is a BOS of type (p, q) with respect to Q.
Note From now onwards we confine our attention to the case of quadratic
forms Q whose B is of maximal rank, since, see section 1.1, these are the only
ones of relevance to our Clifford algebra concerns.
11
We wish to define a function (p, q) of the non-negative integers p, q by
(p, q) = (Q), for any Q of type (p, q) w.r.t. any BOS. (4.2)
(Recall the definition of (Q) {+1, 0, 1} after equation (2.11).) The defini-
tion (4.2) is allowed on account of the following lemma.
Lemma 4.1 If also Q
0
is of type (p, q) with respect to a BOS S
0
=
{a
0
n
, . . . , a
0
n
} for (V
n
, B
0
) (with B
0
of maximal rank), then (Q
0
) = (Q).
Proof If T GL(V
n
) is defined via T a
i
= a
0
i
, then B
0
=
T
B and Q
0
=
T
Q,
whence (Q
0
) = (Q). []
Lemma 4.2 (p + 1, q + 1) = (p, q) (“mod (1,1) periodicity”).
Proof Given (V
n+2
, B), choose a decomposition V
n+2
= V
n
V
2
with V
2
non-singular. Choose BOS’s {b
1
, . . . , b
n
}, {w, v} for V
n
, V
2
and define Q (com-
patible with B) by Q = Q
0
Q
1
, where Q
0
on V
n
is of type (p, q) with re-
spect to {b
1
, . . . , b
n
} and Q
1
on V
2
satisfies Q
1
(w) = 0 and Q
1
(v) = 1. So
Q
1
is of type (1,1); noting that Q
1
(w + v) = 0 + 1 + 1 = 0, observe that
(Q
1
) = +1, and so (1, 1) = +1. Setting a
i
= b
i
+ w + v note, cf. Lemma
3.5, that S {a
1
, . . . , a
2m2
, w, v} is a BOS for V
n+2
, and that Q is of type
(p + 1, q + 1) with respect to S. So in order to prove the lemma we need to show
that (Q) = (Q
0
). To this end consider the 4 cosets V
n
, w+v+V
n
, w+V
n
, v+V
n
of V
n
in V
n+2
. For x V
n
we have Q(w+v +x) = Q
0
(x), Q(w +x) = Q
0
(x), and
Q(v + x) = Q
0
(x) + 1. Consequently the first two cosets each contribute δ(Q
0
)
to δ(Q), while the last two cosets together contribute 0. Hence δ(Q) = 2δ(Q
0
),
whence by equation (2.14), (Q) = (Q
0
). []
Lemma 4.3
(i) (p + 1, q) = (q + 1, p),
(ii) (p, q + 1) = (q, p + 1).
Proof In lemma 2.3 take (i) a = a
i
, (ii) a = a
p+q+1
, where {a
1
, . . . , a
p+q+1
}
denotes a BOS with respect to which Q
0
has type (i) (p + 1, q), (ii) (p, q + 1). []
Corollary 4.4
(i) (p, q + 4) = (p + 4, q) (“mod (4, 4) periodicity”),
(ii) (p + 8, q) = (p, q) = (p, q + 8) (“mod 8 periodicity”).
Theorem 4.5
(p, q) =
+1
0
1
according as p q
0, 1, 2 (mod 8)
3, 7 (mod 8)
4, 5, 6 (mod 8)
.
Proof Let the values (p, q) be set out as a table in which the columns are
labelled by p q and the rows by p + q. By lemma 4.2 the values are constant
12
down each column. By lemma 4.3 the table is symmetric about the column
p q = 1, and skew symmetric about the column p q = 1 (which last is
therefore a column of 0’s). The whole table now follows from the trivial values
(1, 0) = 1, (1, 1) = 1 (this last value being as noted in the course of proving
lemma 4.2) upon repeatedly reflecting about the two columns p q = ±1
leading to the stated values for (p, q). []
Remark In section 5 we shall see that the foregoing derivation of the table
of values of (p, q) is tantamount to a derivation of the table of the real Clifford
algebras Cl(p, q) as given in Porteous (1981). However from the point of view
of finite geometry it is natural to pursue our present concerns a little further in
order to discover precisely which kinds of MOS exist for a given Q. To this end
we will have need of the next result, the computation of which is, as mentioned
in connection with equation (1.6), essentially well-known to Clifford algebraists.
Lemma 4.6 If Q is of type (p, q) w.r.t. the BOS {a
1
, . . . , a
n
}, and u = Σ
i
a
i
,
then
Q(u) =
0
1
according as p q
0, 1 (mod 4)
2, 3 (mod 4)
.
Proof Let a
ij
= a
i
+a
j
, and note that, for example a
12
.a
34
= 1+1 + 1 + 1 = 0
and Q(a
12
) = Q(a
1
) + Q(a
2
) + 1. Consequently, since u = a
12
+ a
34
+ . . . , we
obtain
Q(u) = Σ
i
Q(a
i
) + [n/2] = q + [n/2].
Remark It is now an easy matter to derive finite geometry results which are
somewhat more general than those of Dye (1992): see appendix A.
5 Finite Geometry Derivation of the Table of
Real Clifford Algebras
>From now on the finite geometry of (V
n
, B, Q) will be that arising as in section
1 from the Dirac group G(p, q) Cl(p, q) via the central extension
1 1} G(p, q)
π
V
n
1. (5.1)
If x V
n
then π
1
(x) = {e
x
, e
x
}, where, as in equation (1.3), e : V
n
G(p, q)
denotes a choice of section. In section 6 we will give a dictionary of translation
between concepts and results for (V
n
, B, Q) and their counterparts in G(p, q).
For the present let us just note from (1.5) that we have the basic connection
(B(x, y) =)x.y =
0
1
according as e
x
e
y
=
+e
y
e
x
e
y
e
x
. (5.2)
13
Consequently the notion in section 3 of an OS {a
1
, . . . , a
n
} translates into
that of an AS (=Anticommuting Set) {u
1
, . . . , u
n
} G(p, q), where u
i
u
j
=
u
j
u
i
, i 6= j, and π
1
(a
i
) = {u
i
, u
i
}. Similarly the notions of a BOS, COS,
DOS, MOS translate into corresponding notions, say BAS, CAS, DAS, MAS,
for G(p, q). For example the orthonormal basis {e
1
, . . . , e
n
} for IR
p,q
is a BAS
G(p, q) which corresponds to a BOS for V
n
of type (p, q) with respect to Q;
also, if n is even, {e
1
, . . . , e
n
, w} is a MAS for G(p, q), cf. Theorem 3.6(iii),
where w is the volume element of (1.6).
5.1 Representations Of Dirac Groups: The Frobenius-Schur
Indicator
We first of all need to remind ourselves of those complex irreducible represen-
tations (irreps) of G(p, q), p + q = n, which faithfully represent 1}. The class
structure of G entails that in the even case n = 2m there is one such irrep D, of
dimension 2
m
(the remaining 2
n
irreps of G being the 1-dimensional ones arising
from V
n
) and that in the odd case n = 2m+1 there are two such irreps D
+
, D
,
each of dimension 2
m
. By Schur’s Lemma D
+
(w) = αI, with α = ±1 or ± i
according as w
2
= +1 or 1. D
can be taken to agree with D
+
on the even
elements of G(p, q), but differ by a minus sign on the odd elements; in particular
D
(w) = αI.
Now the complex irreps T of Cl(p, q)
c
= C Cl(p, q), (the complexification
of Cl(p, q)) go hand in hand with those complex irreps D of G(p, q) which
satisfy D(1) = I (with T (g) = D(g) for g G). For T, being linear, must
satisfy T (λ1) = λI, λ C. We have just seen that such a D has dimension
2
m
, where m = [n/2]. Being irreducible, the enveloping algebra of D is, by
Burnside’s theorem, isomorphic to the full matrix algebra M
d
( C), d = 2
m
, of
2
m
× 2
m
complex matrices. The irreps D, D
+
, D
of the preceding paragraph
thus yield algebra surjections T, T
+
, T
: Cl(p, q)
c
M
d
( C). In the even case
n = 2m, a comparison of dimensions shows that T is an isomorphism. In the
odd case n = 2m + 1 the algebra Cl(p, q)
c
possess the central idempotents
e
+
, e
= 1 e
+
, where e
+
= (1 + w)/2, if w
2
= +1, and e
+
= (1 + iw)/2, if
w
2
= 1. So Cl(p, q)
c
is in this case the direct sum I
+
I
of the ideals I
+
, I
generated by e
+
, e
, with T
+
, T
mapping one of these ideals isomorphically
onto M
d
( C) and annihilating the other ideal. So we have the (very well-known)
result:
Cl(p, q)
c
=
M
2
m
(C)
M
2
m
(C)M
2
m
(C)
according as
n = 2m
n = 2m + 1
. (5.3)
However we are interested in the real algebras Cl(p, q), and so in order to
progress further we need to know the reality type of the irrep D of G(p, q),
as given by its Frobenius-Schur indicator η(p, q), which by definition equals
+1, 0 or 1 according as D is of real, strictly complex or quaternionic type,
respectively. Now η(p, q) is determined either (see Shaw (1986)) by
Av
G
D(g
2
) = η(p, q)/ dim D (5.4)
14
or, upon taking the trace, by the well-known classical criterion
Av
G
χ(g
2
) = η(p, q) (χ = character of D). (5.5)
Theorem 5.1 η(p, q) = (p, q).
Proof A (p, q)-orthonormal basis for IR
p,q
yields a BOS with respect to which
Q in (1.4) is of type (p, q), i.e. (Q) = (p, q). Now in computing the left hand
side of (5.4) each x on Q contributes +2, since (e
x
)
2
= 1 = (e
x
)
2
, and each x off
Q contributes 2, since (e
x
)
2
= 1 = (e
x
)
2
. Hence AvD(g
2
) = 2δ(Q)/2
n+1
=
(Q)/2
m
, on using (2.14). The theorem thus follows from (5.4). (In the cases
6= 0 we also recover the fact that dim D = 2
m
.) []
5.2 The Table Of Real Clifford Algebras
Next note that the Frobenius-Schur type (p, q) of the complex irrep D of G(p, q)
determines the real Clifford algebra Cl(p, q), as follows. If n = 2m, and d = 2
m
,
then
Cl(p, q)
=
M
d
(R)
M
d/2
(H)
according as (p, q) =
+1
1
. (5.6)
while if n = 2m + 1, and d = 2
m
, then
Cl(p, q)
=
M
d
(R)M
d
(R)
M
d
(C)
M
d/2
(H)M
d/2
(H)
according as (p, q) =
+1
0
1
. (5.7)
For in the even case we know from theorem 4.5 that (p, q) = ±1, so that
an antilinear operator K exists satisfying KD(g) = D(g)K and K
2
= I. On
restricting the previous algebra isomorphism T to the real algebra Cl(p, q) we
thus obtain (5.6) from (the n = 2m case of) (5.3). In the odd case recall, from
the definition of (Q) prior to lemma 2.2, that = ±1 or = 0 according as
Q(k) = 0 or Q(k) = 1 and therefore according as w
2
= +1 or w
2
= 1. If
= ±1 then the idempotents e
+
, e
lie in the real Clifford algebra, which is
therefore seen to be the direct sum of two ideals each isomorphic to M
d
(IR) (if
= +1) or M
d/2
(IH) (if = 1), d = 2
m
, as in (5.7). The case = 0 is different
in that e
+
= (1 + iw)/2 does not lie in the real algebra. Indeed the central
element w, satisfying w
2
= 1, itself acts as imaginary unit and so Cl(p, q) is
isomorphic to the complexification of the even subalgebra of Cl(p, q), i.e. to
M
d
( C).
Our chief aim has now been achieved, in that we see that the finite geometry
table of (p, q), as provided by theorem 4.5, translates via (5.6) and (5.7) into the
well-known table of real Clifford algebras (see Table 13.26 in Porteous (1981)):
Theorem 5.2 If p + q = 2m, and d = 2
m
, then
Cl(p, q)
=
M
d
(R) if p q 0, 2 (mod 8)
M
d/2
(H) if p q 4, 6 (mod 8)
.
15
If p + q = 2m + 1, and d = 2
m
, then
Cl(p, q)
=
M
d
(R)M
d
(R)
M
d
(C)
M
d/2
(H)M
d/2
(H)
if p q
1 (mod 8)
3, 7 (mod 8)
5 (mod 8)
.
Remark In the proof of theorem 4.5 we made use of the skew symmetry
about the column p q = 1 resulting from lemma 4.3 (ii). In Clifford algebra
terms this means that M
d
(IR) reflects into M
d/2
(IH), M
d
(IR) M
d
(IR) reflects
into M
d/2
(IH) M
d/2
(IH), while M
d
( C) is its own reflection.
5.3 The Graded Real Clifford Algebras
The five kinds of algebra in theorem 5.2 classify the ungraded real Clifford
algebras. As we have shown, they arise from the five kinds of projective quadric
over GF(2), H
2m1
, E
2m1
, Π
0
H
2m1
, P
2m
, Π
0
E
2m1
, whose bilinear form is
of maximal rank. However one ought to classify as well the graded Clifford
algebras, which arise out of the Z
2
-grading Cl(p, q)
0
Cl(p, q)
1
of Cl(p, q),
where Cl(p, q)
0
denotes the even subalgebra (consisting of elements fixed under
the main involution). For example the ungraded algebras Cl(4, 2) and Cl(3, 3)
are isomorphic, both being
=
M
8
(IR), but as graded algebras they are not
isomorphic, since Cl(4, 2)
0
=
M
4
( C) and Cl(3, 3)
0
=
M
4
(IR) M
4
(IR).
Now it is easy to see (by Clifford algebra or finite geometry) that we have
the well-known isomorphisms
Cl(p, q + 1)
0
=
Cl(p, q) and Cl(p + 1, q)
0
=
Cl(q, p). (5.8)
Upon using theorem 5.2 one then easily sees that there are eight kinds of graded
real Clifford algebras. It is straightforward to give a finite geometry interpre-
tation of these eight kinds. For now V
n
comes along with a preferred “even
hyperplane” H, whose equation is
P
i
x
i
= 0 in coordinates with respect to a
(p, q)-BOS {a
1
, . . . , a
n
}. Corresponding to the Dirac group G(p, q) possessing
a privileged subgroup of index 2, namely the even Dirac group
G(p, q)
0
= G(p, q) Cl(p, q)
0
, (5.9)
the quadric Q has a privileged “subquadric” Q
0
, the restriction of Q to the
hyperplane H. For a quadric Q of the three types (i) H
2m1
, (ii) E
2m1
, (iii)
P
2m
, there are two types of section Q
0
by H, namely (i) Π
0
H
2m3
or P
2m2
,
(ii) Π
0
E
2m3
or P
2m2
, (iii) H
2m1
or E
2m1
, respectively. (In the last case the
section can not be of type Π
0
P
2m2
, since the nucleus k =
P
a
i
of P
2m
does not
lie in H.) On the other hand for Q of the remaining two types (iv) Π
0
H
2m1
, (v)
Π
0
E
2m1
, there is only one type of section Q
0
, namely (iv) H
2m1
, (v) E
2m1
,
since H does not pass through the vertex k of these cones. So we see that there
are indeed precisely eight kinds of possibilities for the pair Q
0
, Q.
16
6 Finite Geometry and the Dirac Group I: Sub-
spaces and Subgroups
6.1 A Dictionary Of Translation
Of course the subgroups of V
n
are precisely the vector subspaces X V
n
(and
are normal subgroups since V
n
is abelian). We thus have from (5.1), see e.g.
Hall (1976), theorem 2.3.4, a 1 - 1 correspondence between subspaces X of V
n
and those subgroups (necessarily normal) of G(p, q) which contain 1. Let the
subgroup π
1
(X) of G(p, q) corresponding to the subspace X of V
n
be denoted
G(X). Note that X has dimension r if and only if G(X) has order 2
r+1
. Observe
that we have a 1 - 1 correspondence between complete flags
{0} = V
0
V
1
. . . V
n
(6.1)
of subspaces of V
n
(with dim V
r
= r) and composition series (missing {1}) for
G(p, q) of the type
(1 )1} = G
0
G
1
. . . G
n
= G(p, q) (6.2)
where |G
r
| = 2
r+1
(each G
r
being necessarily normal in G
n
, since in (6.2) the
subgroups G
r
all contain 1}).
Let C(H) denote the centralizer in G(p, q) of a subgroup H. Then clearly
Y = X
if and only if G(Y ) = C(G(X)). (6.3)
In particular the radical (V
n
)
of V
n
corresponds to the centre of G(p, q). From
(6.3) we also see that X being (totally) isotropic corresponds to G(X) being
abelian:
X X
if and only if G(X)(3 1) is abelian. (6.4)
Consequently the maximal isotropic subspaces of V
n
are in 1 - 1 correspondence
with the maximal abelian normal subgroups of G(p, q). In the even case n = 2m
the latter thus have order 2
m+1
, while in the odd case n = 2m + 1 they have
order 2
m+2
.
It is also immediate that direct sum decompositions X Y of V
n
are in 1
1 correspondence with factorizations of G(p, q) as a product HK of normal
subgroups H = G(X) and K = G(Y ) such that H K = {1, 1}. In particular
V
n
= X Y if and only if G(p, q) = G(X) G(Y ), (6.5)
where H K denotes the central product over 1} of mutually centralizing
normal subgroups whose intersection is 1}. (This is not their direct product
H ×K, for which (h, k) is distinct from (h, k) in contrast to h(k) = (h)k
H K.)
Remark Parts of the dictionary are simpler in the even case n = 2m. For then
B is non-degenerate and every hyperplane X is of the form X = x
for
17
unique non- zero x V. It follows that every subgroup of index 2 in G(p, q), p +
q = 2m, is the centralizer C(g) = C(g) of (either of) a unique pair of elements
{g, g} 6= {1, 1}. Incidentally C(g) is always itself a Dirac group (for the odd
dimension 2m 1), since the restriction of B to a hyperplane has 1-dimensional
kernel, i.e. maximal rank. In particular C(w) is the even Dirac group G(p, q)
0
,
see (5.9). If we define x
0
= {y V
2m
: y.x = 1}, then corresponding to the
decomposition V
2m
= x
x
0
we have the decomposition G(p, q) =
C(e
x
) C
(e
x
), where C
(g) denotes the anticentralizer of g, consisting of
all those elements of G(p, q) that anticommute with g. Note that the choice
e
x
= w yields the decomposition G(p, q) = G(p, q)
0
G(p, q)
1
of the Dirac
group arising from the Z
2
-grading of Cl(p, q). In the even cases note also that
every non-trivial normal subgroup H of G(p, q) is of the form G(X) for some
unique subspace X of V
n
for such H necessarily contains 1}. (In the case
of odd n, with signature such that, see (1.6), w
2
= +1, note that {1, w} is a
normal subgroup which does not contain 1}.) In the even cases the Dirac
groups G(p, q) are extraspecial 2-groups, see e.g., Frenkel et al. (1988), Braden
(1985), with structure as described in (7.37).
6.2 Subspaces And Subgroups: dim(X) 3
In section 6.1 we used only the scalar product (5.2). However we need to know
how a subspace X V
n
relates to the quadratic form Q in order to determine the
nature of the corresponding subgroup G(X) of the Dirac group. Consider first
of all the case when X = x is 1-dimensional, the corresponding subgroup
G(X) = 1, ±e
x
} being of order 4. Clearly G(X) is isomorphic to (Z
2
)
2
, or
to Z
4
, according as the point x lies on, or off, the quadric Q.
Secondly consider the cases when X has dimension 2, and so G(X) is of
order 8. Now up to isomorphism there are 5 groups of order 8, namely
Z
8
, Z
4
× Z
2
, (Z
2
)
3
, D
8
, Q
8
. (6.6)
Of course Z
8
does not occur as a G(X), since every element of G(p, q) has order
1, 2 or 4, but, as we now show, the remaining 4 groups in (6.6) all arise as a
G(X). As far as B is concerned there are two possibilities for X :
(i) X is an isotropic plane (and so G(X) is abelian)
(ii) X is non-singular, i.e. a hyperbolic plane (so G(X) is non-abelian).
(6.7)
Let us from now on adopt projective language. In case (i) the isotropic (self-
polar) line λ = IPX may either
(i
a
) lie on Q, or
(i
b
) be tangent to Q, i.e. intersect Q at a single point. (6.8)
18
(Setting B = 0 in (2.1) the only possibilities for the 3 points x, y, x + y on the
projective line λ are that Q = 1 for (a) 0 or (b) 2 of these points.) Accordingly
G(X) has
(a) 3 subgroups
=
(Z
2
)
2
(b) 1 subgroup
=
(Z
2
)
2
and 2 subgroups
=
Z
4
.
Consequently we have
(i
a
) G(X)
=
(Z
2
)
3
(i
b
) G(X)
=
Z
4
× Z
2
. (6.9)
In case (ii) we see similarly that the hyperbolic line λ for which B(x, y) = 1
for distinct points x, y λ is either
(ii
a
) a bisecant to Q, i.e. 2 points on, and 1 off, Q, or
(ii
b
) an exterior line, i.e. all 3 points lie off Q. (6.10)
Accordingly G(X) has
(a) 2 subgroups
=
(Z
2
)
2
and 1 subgroup
=
Z
4
(b) 3 subgroups
=
Z
4
.
Consequently
(ii
a
) G(X)
=
D
8
(ii
b
) G(X)
=
Q
8
. (6.11)
(These latter are the two types of Dirac group in dimension n = 2, see (7.8).)
Let us also briefly consider the cases when X has dimension 3. Then the
restriction of B to X may either (i) be zero, i.e. have rank 0, or (ii) have rank
2. In case (i) it follows from (6.8) that there are just two ways in which the
isotropic projective plane α = IPX can relate to the projective quadric Q : either
(i
a
) α lies on Q, or (i
b
) α intersects Q in a line. Accordingly we have
(i
a
) G(X)
=
(Z
2
)
4
,
(i
b
) G(X)
=
Z
4
× (Z
2
)
2
. (6.12)
In case (ii) there are three possibilities, according as α intersects the quadric in
(ii
a
) a Π
0
H
1
, i.e. a pair of lines,
(ii
b
) a P
2
, i.e. 3 non-collinear points,
(ii
c
) a Π
0
E
1
, i.e. a single point. (6.13)
(These 3 possibilities go along with 3 kinds of Dirac group, see (7.9), (7.10).)
19
7 Finite Geometry and the Dirac Group II: Il-
lustrations with n 4
7.1 Introduction
In sections 7 and 8 we will provide further low-dimensional illustrations of the
foregoing dictionary — but first of all it may help to give a brief reminder of our
general results. In even dimension n = 2m the two types, hyperbolic ( = +1)
and elliptic ( = 1), of non-singular quadratic form Q on V
2m
, see (1.4),
are associated, see (5.6), with two corresponding types of simple real Clifford
algebras, isomorphic respectively to M
d
(IR) and to M
d/2
(IH), d = 2
m
. In odd
dimension n = 2m + 1 there is one type, parabolic ( = 0), of non-singular
Q on V
2m+1
, the associated type of real Clifford algebra being isomorphic to
M
d
( C), and so also simple. In this odd-dimensional case there are also two
types of singular Q such that the associated alternating form B is of rank 2m,
namely cones of hyperbolic ( = +1) and elliptic ( = 1) type, associated,
see (5.7), with the two types of real Clifford algebras which are the direct sum
of two simple algebras, being isomorphic respectively to M
d
(IR) M
d
(IR) and
M
d/2
(IH) M
d/2
(IH), d = 2
m
.
For a given signature (p, q) the isomorphism type of the algebra Cl(p, q) may
of course be obtained from theorem 5.2. Alternatively in low dimension it may
just as easily be read off from the lists (A.6), (A.7) of MOS’s in appendix A.
In the even cases all we need to do is to consult (A.6), and recall that BOS’s
are derived from MOS’s by removing one (any) point. In the case n = 2, for
example, we see from (A.6) that for a hyperbolic quadric a MOS is of type (2,1)
only, leading to BOS’s only of types (2,0) and (1,1), and similarly we see that
for an elliptic quadric a BOS must be of type (0,2). Consequently we have the
isomorphisms
( = +1) Cl(2, 0)
=
Cl(1, 1)
=
M
2
(R)
( = 1) Cl(0, 2)
=
H. (7.1)
Similarly in the cases n = 4 and n = 6 we can quickly read off from the second
and third rows of (A.6) the isomorphisms
( = +1) Cl(3, 1)
=
Cl(2, 2)
=
M
4
(R)
( = 1) Cl(4, 0)
=
Cl(1, 3)
=
Cl(0, 4)
=
M
2
(H) (7.2)
20
and
( = +1) Cl(4, 2)
=
Cl(3, 3)
=
Cl(0, 6)
=
M
8
(R)
( = 1) Cl(6, 0)
=
Cl(5, 1)
=
Cl(2, 4)
=
Cl(1, 5)
=
M
4
(H). (7.3)
In the case of odd dimension, we can immediately read off = 0 results from
(A.7), with the first two rows yielding the isomorphisms
Cl(3, 0)
=
Cl(1, 2)
=
M
2
( C) (7.4)
Cl(4, 1)
=
Cl(2, 3)
=
Cl(0, 5)
=
M
4
( C). (7.5)
Bearing in mind what was said after (A.4), we can also use (A.6) as a convenient
source of information for the remaining odd-dimensional cases. Thus from the
first two rows of (A.6) we can read off the isomorphisms
( = +1) Cl(2, 1)
=
M
2
(R) M
2
(R)
( = 1) Cl(0, 3)
=
H H (7.6)
and
( = +1) Cl(3, 2)
=
M
4
(R) M
4
(R)
( = 1) Cl(5, 0)
=
Cl(1, 4)
=
M
2
(H) M
2
(H). (7.7)
In even dimension n = 2m the two types of real Clifford algebras will give rise
to two types of Dirac groups, say G
(2m), = ±1. In odd dimension n = 2m+1
there will similarly be three types of Dirac group, say G
(2m+1), = +1, 0, 1.
(Caution: we are here neglecting the gradation aspects! see section 5.3.)
Now, using the dictionary in section 6, we may use finite geometry to illuminate
properties of these Dirac groups, and this we will do in the following sections 7.2
7.4 and 8. In fact we will concentrate upon certain low-dimensional illustra-
tions, and we wish to deal especially with (some of!) the Dirac group/Clifford
algebra spin-offs from the finite geometry symmetry ideas outlined in appendices
B D.
7.2 Dimensions n 4
In dimension n = 2 the two types of Dirac group G
(2) arise from the two types
of hyperbolic line, as in (6.10), the projective quadric either being a H
1
, having
two points, or an E
1
, having no points. So from (6.11), or otherwise, we have
(a) G
+
(2)
=
G(2, 0)
=
G(1, 1)
=
D
8
21
(b) G
(2)
=
G(0, 2)
=
Q
8
. (7.8)
In dimension n = 3 the three types of Dirac group G
(3) arise from the three
types of projective quadric listed in (6.13), namely Π
0
H
1
, P
2
, Π
0
E
1
, for =
+1, 0, 1. One easily sees that for ± 1 G
(3) is isomorphic to G
(3)
0
× Z
2
:
G
+
(3)
=
G(2, 1)
=
D
8
× Z
2
and G
(3)
=
G(0, 3)
=
Q
8
× Z
2
, (7.9)
with Z
2
= {1, w}, or equally Z
2
= {1, w}, while for = 0 we have, e.g. using
(6.5),
G
0
(3)
=
G(3, 0)
=
G(1, 2)
=
D
8
Z
4
=
Q
8
Z
4
, (7.10)
with Z
4
= hwi = 1, ±w} and with the D
8
arising as G(1, 2)
0
and the Q
8
as
G(3, 0)
0
. Use of (6.5) would also yield G
+
(3)
=
D
8
(Z
2
)
2
and G
(3)
=
Q
8
(Z
2
)
2
, but one easily sees that there is no conflict with (7.9), since D
8
(Z
2
)
2
=
D
8
× Z
2
and Q
8
(Z
2
)
2
=
Q
8
× Z
2
.
It is an easy matter to list the maximal subgroups of these n = 3 Dirac
groups. For example, a conic P
2
in PG(2, 2) possesses 3 tangents (passing
through its nucleus), 3 bisecants and one exterior line. Accordingly G
0
(3) pos-
sesses 3 subgroups
=
Z
4
× Z
2
, 3 subgroups
=
D
8
and one subgroup
=
Q
8
. Of
course in such a low dimension as n = 3 there is no need to use finite geometry
to obtain such results, as they can easily be obtained in a more orthodox fash-
ion. In dimensions n 4, however, use of finite geometry is a great boon, as we
now proceed to demonstrate. We will treat n = 4 immediately, and n = 5, 6 in
section 8.
Let us therefore now look at an algebra A(4) = Cl(p, q), p + q = 4, and
associated Clifford group G(4) (of order 32). Such an algebra is of dimension
16, and a customary treatment would display a basis for it in the form
{1, e
i
, e
i
e
j
, e
5
e
i
, e
5
}, (i < j), (7.11)
with {e
1
, e
2
, e
3
, e
4
} an orthonormal basis for IR
p,q
, p+q = 4, and e
5
= e
1
e
2
e
3
e
4
.
However, as already announced, we wish to demonstrate how to take advantage
of the finite geometry symmetry methods of appendices B D. So, taking our
inspiration from appendices B and C, let us start out in 2 dimensions higher with
a Clifford algebra A(6) = Cl(r, s), r + s = 6, generated by an anticommuting
set
S = {u
1
, . . . , u
6
}, with u
i
2
=
i
= ±1. (7.12)
Consider the even subalgebra A(5) = A(6)
0
of dimension 32 (= 1+15+15+ 1) :
A(5) = 1, u
ij
, wu
ij
, w , (7.13)
where u
ij
= u
i
u
j
, i 6= j, and w = u
1
u
2
u
3
u
4
u
5
u
6
, and consider the associated
even Dirac group G(5) = G(6)
0
, of order 64. Of course, cf. (B.7), each of the 6
sets
B
i
= {u
ij
: j 6= i} (7.14)
22
is a BAS for G(5), i.e. a set of 5 anticommuting elements which generate G(5).
We now choose s to be odd, so as to achieve w
2
= +1. Then e
+
1
2
(1 + w) is a
central idempotent for A(5) which therefore generates an ideal
e
+
A(5) = e
+
, e
+
u
ij
(7.15)
of dimension 16 (= 1 + 15). In its own right this ideal is an algebra A(4), with
e
+
as 1. Setting, for i 6= j,
γ
ij
= e
+
u
ij
= e
+
(u
i
u
j
), (7.16)
cf. (B.9), we see that we may take the algebra A(4) of (7.11), and associated
Dirac group G(4), in the forms
A(4) = 1, γ
ij
, G(4) = 1, ±γ
ij
}. (7.17)
(An alternative view of A(4) is possible, and may be preferred: one may think
of A(4) as arising from an irreducible representation u
ij
γ
ij
of A(5) such that
w I.) Of course under the projection π of (5.1), G(4) maps onto the sym-
plectic space V
4
= {0, p
ij
} of appendix C, where the points of IPV
4
= PG(3, 2)
are the 15 p
ij
= π(±γ
ij
} = p
ji
. Of course, cf. (B.11), the 6 BAS’s for G(5) in
(7.14) give rise to the following 6 MAS’s (i.e. maximal-size anticommuting sets)
for G(4) :
M
i
= {γ
ij
: j 6= i}. (7.18)
Observe that corresponding to the symplectic geometry relations (B.10), we
have of course the commutativity/anticommutativity properties
γ
ij
γ
ik
= γ
ik
γ
ij
and γ
ij
γ
kl
= +γ
kl
γ
ij
. (7.19)
¿From (7.12) and (7.16) the γ
ij
also satisfy the relations
γ
ij
= γ
ji
γ
ij
2
=
i
j
γ
ij
γ
jk
=
j
γ
ik
, (7.20)
and, on account of u
1
u
2
u
3
u
4
u
5
u
6
= w, the relations
γ
ij
γ
kl
γ
mn
= η
ijklmn
. (7.21)
As in appendix C, we here let ijklmn denote an arbitrary permutation of 123456,
but we also here let η
ijklmn
denote the sign of this permutation.
It now immediately follows from (iii), (iv) of section C.2 that a Dirac group
G(4) possesses precisely 35 normal subgroups of order 8, classified as follows:
15 maximal abelian normal subgroups G(ij, kl, mn), where
G(ij, kl, mn) = 1, ± γ
ij
, ± γ
kl
, ± γ
mn
}; (7.22)
20 non-abelian normal subgroups G
ijk
, where
G
ijk
= 1, ± γ
ij
, ± γ
jk
, ± γ
ki
}, (7.23)
and where, see (6.3), these 20 subgroups form 10 mutually centralizing
pairs {G, C(G)}, with C(G
ijk
) = G
lmn
.
23
Observe that so far we have only used the Sp(4, 2)-geometry of the space V
4
,
and so the results (7.22), (7.23) apply to both types G
(4), = ±1, of Dirac
groups! To progress further we need to feed in the type of O
(4, 2)-geometry on
V
4
relevant to the particular algebra A(4) = Cl(p, q), as given by (7.2).
7.3 The G
(4) Dirac Groups
Consider first of all the algebra Cl(4, 0) of “elliptic” type = 1. We can
achieve this in (7.17) by choosing
i
=
6
, i 6= 6, i.e. by starting out from
A(6) = Cl(5, 1) or = Cl(1, 5). For then we have
γ
i6
2
= +1 and γ
ij
2
= 1, i, j 6= 6, (7.24)
and so can tie in (7.17) with (7.11) by, for example, writing
e
i
= γ
i6
, i = 1, . . . , 5. (7.25)
Consequently in PG(3, 2) = IPV
4
we now have a preferred elliptic quadric E
3
whose points are the 5 p
i6
, i 6= 6. In conformity with (A.6), note that while all
5 elements of M
6
have squares = +1, just one element of M
i
, i 6= 6, has square
= +1. By using (6.9) and (6.11), we can immediately read off from (viiib) of
section C.2 the following features of the Dirac group G(4, 0) :
each of the 15 maximal abelian normal subgroups G(ij, kl, mn) is
isomorphic to Z
4
× Z
2
; (7.26)
each of the 10 subgroups G
lm6
is isomorphic to D
8
, and each of
their 10 centralizers G
ijk
= C(G
lm6
) is isomorphic to Q
8
. (7.27)
In fact any geometrical result for an elliptic quadric E
3
in PG(3, 2) will have its
group theory counterpart for a Dirac group G
(4). For example, from (viiia)
of section C.2 we can deduce the isomorphisms C(γ
i6
)
=
G
(3) and C(γ
ij
)
=
G
0
(3), for i, j 6= 6. (Also, since the line conjugate to a bisecant is an external
line, the centralizer of a D
8
is always a Q
8
, as in (7.27).) As another example,
through each point (say p
lm
) lying off E
3
there passes a unique bisecant (namely
λ
lm6
); consequently each Z
4
subgroup of G
(4), such as 1, ±γ
lm
}, lies inside
a unique D
8
subgroup, namely G
lm6
. One sees similarly that a Z
4
subgroup of
G
(4) always lies inside 6 other subgroups of order 8, namely 3 Z
4
× Z
0
2
s and
3 Q
0
8
s.
The foregoing applies equally to the other G
(4) Dirac groups G(1, 3) and
G(0, 4). To link up with G(1, 3) we may in (7.11) set e
1
= γ
61
and e
i
= γ
1i
, i =
2, 3, 4, 5, and to link up with G(0, 4) we may set e
i
= γ
i5
, i = 1, 2, 3, 4 and
e
5
= γ
56
.
7.4 The G
+
(4) Dirac Groups
Next let us consider the “Majorana algebra” Cl(3, 1)
=
M
4
(IR), whose Dirac
group G(3, 1) is of hyperbolic type G
+
(4). To achieve this signature for A(4)
24
in (7.17) we need to start out from neutral signature (3,3) for A(6), and so in
(7.12) we will set
1
=
2
=
3
= +1 and
4
=
5
=
6
= 1. Then precisely 9 of
the γ
ij
have squares equal to +1 :
γ
ab
2
= +1, a {1, 2, 3}, b {4, 5, 6}, (7.28)
and to tie in with (7.11) we could set
e
i
= γ
i6
, i = 1, 2, 3, 4, e
5
= γ
56
. (7.29)
Consequently in PG(3, 2) = IPV
4
we now have a preferred hyperbolic quadric
H
3
, with points the 9 p
ab
, a {1, 2, 3}, b {4, 5, 6}, which was denoted H
123
=
H
456
in (C.4). In conformity with (A.6), note that each of the 6 maximal
anticommuting sets (7.18) includes 3 elements with squares +1 and 2 elements
with squares 1.
¿From (ixb) of section C.2 we can read off the following facts concerning
those normal subgroups (7.22), (7.23) of the Dirac group G(3, 1) which have
order 8:
6 of the maximal abelian normal subgroups are isomorphic
to (Z
2
)
3
and 9 are isomorphic to Z
4
× Z
2
; (7.30)
there are precisely 2 normal subgroups isomorphic to the
quaternion group Q
8
, namely G
123
and G
456
, and these centralize
one another, and the remaining 18 subgroups G
ijk
, forming 9
mutually centralizing pairs, are all isomorphic to D
8
. (7.31)
Also, from (ixa) of section C.2, we may deduce that C(γ
ij
) is isomorphic to
G
+
(3) or to G
0
(3) according as γ
ij
is as in (7.28), with square = +1, or has
square = 1.
We can say more concerning those 6 maximal abelian normal subgroups of
G(3, 1), out of the 15 G(ij, kl, mn) in (7.22), which are isomorphic to (Z
2
)
3
.
Since two normal subgroups of a Dirac group G(n), n > 1, always have at least
±1 in common, let us say that they are disjoint if they intersect only in 1}.
Otherwise stated, G(X) and G(Y ) are disjoint whenever X Y = {0}, i.e.
whenever IPX is skew to IPY. It follows from (C.5) that the 6 normal subgroups
of G(3, 1) which are isomorphic to (Z
2
)
3
form two systems of three mutually
disjoint subgroups: the subgroups of one system have labels ij, kl, mn as given
by the three rows, and of the other system by the three columns, of the array
14 25 36
35 16 24
26 34 15,
(7.32)
a subgroup of one system thus intersecting a subgroup of the other system
in a (Z
2
)
2
subgroup of the the form 1, ±γ
ab
}. In a 4-dimensional notation
appropriate to Cl(3, 1) this array could read
25
14 134 3
124 1 24
2 34 234.
(7.33)
Thus the second row of the array labels the (Z
2
)
3
subgroup
G(35, 16, 24) = 1, ±γ
35
, ±γ
16
, ±γ
24
} = 1, ±e
1
e
2
e
4
, ±e
1
, ±e
2
e
4
}.
(7.34)
The maximal abelian normal subgroups of type (Z
2
)
3
are of importance for the
following reason. If H = 1, ±g
1
, ±g
2
, ±g
3
} is one of these subgroups, then
the 4 products
1
2
(1±g
1
)
1
2
(1±g
2
) form a set of 4 mutually annihilating primitive
idempotents for the Clifford algebra, their sum being 1. (Such idempotents give
rise to a decomposition of the algebra into a direct sum of minimal left ideals,
and are useful in the construction of spinor bases, see e.g. Lounesto & Wene
(1987), Crumeyrolle (1990).) If we choose the signs so that g
1
g
2
g
3
= +1, rather
than 1, then the 4 primitive idempotents defined by H are of the form
1
4
(1 ± g
1
± g
2
± g
3
) (with an even number of minuses). (7.35)
For example, Lounesto & Wene (1987) used the second row of the array (7.33),
as did Benn & Tucker (1987), eq. (2.2.11), while Chisholm & Farwell (1992)
used the second column. (Curiously the arrangement g
1
g
2
g
3
= +1 can be
simultaneously achieved in 5, but not all 6, of the 6 rows and columns.)
We can deal with the other G
+
(4) Dirac group G(2, 2) in a similar way. It
may help if we label the anticommuting set (7.12) for Cl(3, 3) so that
1
=
2
=
5
= +1 and
3
=
4
=
6
= 1, for then we may in (7.11) set e
i
= γ
i6
, i 6= 6
but subsequently we have to interchange the labels 3 and 5, in particular in
(7.28) and (7.32).
Remark Central product structure of Dirac groups Using (6.5), we
deduce from (7.27), (7.31) the following central product structures for the G
(4)
groups:
G
(4)
=
D
8
Q
8
(in 10 ways)
G
+
(4)
=
Q
8
Q
8
(in one way)
=
D
8
D
8
(in 9 ways).
(7.36)
In fact these results easily generalize, and we find
(i) G
+
(2m) is isomorphic to a central product of m copies of D
8
,
(ii) G
(2m) is isomorphic to a central product of m 1 copies
of D
8
along with one copy of Q
8
.
(7.37)
To prove these results we may adopt an inductive argument which uses the
fact that G
(2m + 2)
=
G
(2m) D
8
. This last follows from (6.5) applied to
an appropriate decomposition V
2m+2
= V
2m
V
2
(with IPV
2
a bisecant of the
projective quadric and so, as in (6.11
b
), giving rise to the D
8
). The decompo-
sitions (7.37) of the Dirac groups G
(2m) correspond to those decompositions
26
of V
2m
into an orthogonal sum V
2
. . . V
2
of non-singular planes which
give rise to the canonical forms (2.10
) for Q
. Of course from the isomorphism
D
8
D
8
=
Q
8
Q
8
in (7.36) we may in (7.37i, ii) replace an even number of D
8
factors by the same number of Q
8
’s.
In the case of odd dimension the space V
2m+1
has a one-dimensional radical
k and we may consider decompositions of the kind V
2m+1
= V
2m
k .
In the parabolic case = 0, with k the nucleus of a P
2m
, application of (6.5)
yields
G
0
(2m + 1)
=
G
+
(2m) Z
4
=
G
(2m) Z
4
(7.38)
with Z
4
= hwi. Here G
+
(2m) and G
(2m) arise because, for a hyperplane IPV
2m
not passing through the nucleus, there are two kinds of section, hyperbolic and
elliptic. In the cases G
(2m + 1) with = ±1 we have the simpler results
G
(2m + 1)
=
G
(2m) × Z
2
, = ±1, (7.39)
since the central Z
2
subgroup hwi, or equally h−wi, has trivial intersection with,
e.g., G
(2m + 1)
0
=
G
(2m).
8 Finite Geometry and the Dirac Group III:
Dimensions n = 5 and n = 6
In the preceding section we were able to introduce a 6-index labelling into our
treatment of the n = 4 Dirac groups, in effect by taking advantage of the
isomorphism Sp(4, 2)
=
S
6
. Nevertheless, see after (7.23), once we treat a specific
one of the two types G
(4) of group then the S
6
symmetry is broken, to (S
3
×
S
3
).Z
2
, see (C.6), in the case = +1, and to S
5
, see (B.3
0
), in the case = 1.
However in the case of the Dirac group G
0
(5) we are in the fortunate position
of being able to take full advantage of S
6
symmetry, since, see (B.5
0
), we have
the isomorphism O(5, 2)
=
S
6
. For details of the O(5, 2)-geometry of the Dirac
group G
0
(5), consult Shaw (1994), sections D.1 and 8.1.
In the case of the Dirac group G(0, 6)
=
G
+
(6) we are again in a fortunate
position, since we are able to take advantage of the S
8
symmetry arising from
the isomorphism O
+
(6, 2)
=
S
8
of (B.2
0
). Consider therefore the Clifford algebra
Cl(0, 6) = 1, e
i
, e
i
e
j
, e
i
e
j
e
k
, e
7
e
i
e
j
, e
7
e
i
, e
7
, (8.1)
where the e
i
, i = 1, . . . , 6, are 6 independent anticommuting elements which
satisfy e
i
2
= 1. Their product e
7
e
1
e
2
e
3
e
4
e
5
e
6
also satisfies e
7
2
= 1, and so
we have a heptad {e
1
, . . . , e
7
} of mutually anticommuting imaginary units such
that e
1
e
2
. . . e
7
= 1, such a heptad being the Clifford algebra analogue of what
finite geometers refer to as a Conwell heptad for a H
5
quadric, see appendices
A and D. One is therefore tempted to re-write (8.1) using a 7-index notation, in
effect viewing Cl(0, 6) as one of the ideals of Cl(0, 7) (or non-universal algebra
for IR
0,7
).
27