on combinatorial and projective geometry

ANDREAS W . M . DRESS AND WALTER WENZEL

O N COMBINATORIAL AND PROJECTIVE GEOMETRY*

ABSTRACT. Cross ratios constitute an important tool in classical projective geometry. Using the theory of Tutte groups as discussed in [6] it will be shown in this note that the concept of cross ratios extends naturally to combinatorial geometries or matroids. From a thorough study of these cross ratios which, among other observations, includes a new matroid theoretic version and proof of the Pappos theorem, it will be deduced that for any projective space M = P,(K) of dimension n 3 2 of M over some skewfield K the inner 7iitte group is isomorphic to the commutator factor group K*/[K*,K*] of K*:= K\{O). This shows not only that in case M = P,(K) our matroidal cross ratios are nothing but the classical ones. It can also be used to correlate orientations of the matroid M = P,(K) with the orderings of K. And it implies that Dieudonnt's (non-commutative) determinants which, by Dieudonnt's definition, take their values in K*/[K*, K*] as well, can be viewed as a special case of a determinant construction which works for just every combinatorial geometry.

In 163, for every matroid M a certain abelian group U,, the Tutte group of M , was constructed and studied. It was shown there that an aIgebraic approach to matroid theory, and in particular to the theory of matroids with coefficients, can be based on this construction. By definition, it contains a certain subgroup Tg), called the inner Tutte group of M, and a specified element E M € UE) satisfying EL = 1. It is easy to compute the structure of T, once the structure of T'# is known. All this will be recalled in Section 1.

In Section 2 we start with the observation that for any four hyperplanes HI, H,, H,, H, in M with Hi # H j for i E {1,2) and j E {3,4) which intersect

in a hyperline one can associate a certain element [:: E Tg) which has

essentially all the pleasant properties, cross ratios are known to have in classical projective geometry like, e.g., invariance with respect to appropriately defined projective equivalence. Together with elementary results concerning the Tutte group 8, as established in [6] this yields-among other results-a matroid theoretic version and proof of the Pappos theorem. Moreover, from the more elementary part of Tutte's homotopy theory for matroids it follows that TE) is generated by E, and all these cross ratios. (See [I31 and [15].)

*Research supported by the DFG (Deutsche Forschungsgemeinschaft).

Geometriae Dedicata 34: 161-197, 1990. 1990 Kluwer Academic Publishers. Printed in the Netherlands.

162 A N D R E A S W , M. DRESS A N D W A L T E R W E N Z E L

Finally, still in Section 2, we shall show that cross ratios can be defined more generally for modular quadruples P1, P2, Q1, Q2 of subspaces of M, that is, for subspaces for which the rank satisfies the relations P(PO =

P(P2), P(Q1) = p(Q2), p(Pi) + p(Qj) = p(P, n Qj) + p(P, u Q~), p(P, c~ Qj) =

p(P~ c~ P2 n Q1 n Q2), and P(Pi u Qj) = p (e l u P2 ~ Q1 w Q2) for all i, j e { 1, 2} and we shall discuss harmonic quadruples in this context.

Using these facts we shall compute the Tutte group of a projective space M = P,(K) of dimension at least 2 over some skewfield K in Section 3 and show that the inner Tutte group .~-r<°~ is isomorphic to the abelianized multiplicative group K*/[K* , K*] of K.

Actually, it was this result which suggested to define and study cross ratios, taking their values in Tutte groups of matroids, quite generally. Once established, our approach has many further interesting consequences. We shall see, for instance, that in the case where P~, P2, Pa, P4 are four pairwise distinct points on some line, and therefore constitute a modular quadruple,

[ P21 corresponds exactly to the cross ratio studied in the cross ratio P~ Ps P4

classical projective geometry (see, e.g., [2, Ch. III, §4]) and that as a simple consequence of a result by Grundh6fer (see I-9, Th. 2-1) for any finite non- Desarguesian projective plane M we have T~ ) = {1}. Next, in Section 4 we shall correlate the chirotope structures or 'orientations' of a projective space M over K of dimension at least 2 with the orderings of K. Finally in Section 5, a determinant in the form of a certain coset in T u will be defined for every automorphism ~t of an arbitrary matroid M, and it will be seen that this determinant may be interpreted as Dieudonnffs determinant when M is a vector space over some skewfield K of dimension at least 3 and ct is a linear automorphism of M.

1. P R E L I M I N A R I E S

Let M denote a matroid defined on a possibly infinite set E of finite rank n. Let ~ -- ~ M denote the set of its hyperplanes, p = PM its rank function, and a = aM its closure operator.

In [6, §l] we demonstrated various ways to define the Tutte group of the matroid M. In this paper we shall work with the extended Tutte group and the inner Tutte group whose definitions we repeat for the convenience of the reader.

DEFINITION 1.1. Let U:~ denote the free abelian group generated by the symbols e and XH,a for H e ~ and a e E \ H . Furthermore, let K ~ denote the

O N C O M B I N A T O R I A L A N D P R O J E C T I V E G E O M E T R Y 163

subgroup of ~ generated by /~2 and all elements of the form

e ' X H , ~ ' X -1 . -1 . . X - 1 , H,,aa'XH2,a3 XH2,al XHa,al Ha,a2

for H1, H2, H 3~J~, L = H lc~H 2c~H 3 = H i n H # for l ~ < i < j ~ < 3 , p(L) = n -- 2 and ai E H i \ L for i ~ {1, 2, 3}. Then the extended Tutte group Y ~

of M is defined by

-[]-M~: = .s6 j~ I=M/P~M.

Let #: D=~--~ T ~ denote the canonical epimorphism and put

(1.1a) eM:= #(e);

(1.1b) H(a) = HM(a):= #(Xma) for H~JI6, a ~E \H .

REMARK. In [-6] the elements H(a) were denoted by T~,a. In this paper we shall always use the notation H(a) instead of Tu, a when there is no doubt which matroid M is considered; otherwise we write H~(a).

DEFINITION 12. Let ~: 1-~ ~ 7/~ x Z E denote the obviously well-defined homomorphism given by

~(~M) := 0,

• (n(a)) := (Jn: ~ ~ 7/: n'~--~ 6n n', 6a: E--+ 7/: a' ~-+ J~')

for H ~ ~ , a ~ E\H. Then -r¢o) UM := ker • is called the inner Tutte #roup of M.

REMARK. The Tutte group TM, defined in I-6, Def. 1.1] coincides with the subgroup of TM ~, generated by eu and all elements of the form H(a).H(b)- ~ with H E ~ and a, b ~ E\H; that is, T M is the kernel of the homomorphism ~ e : TM ~__. ZYe given by ~ (eM) := 0, CP:~(H(a)) := Ju. So we have ~-~) ___ TM c qF~.

To compare Tutte groups of different matroids we need

PROPOSITION 1.3. Assume D is a subset of E and E' ~_ E\D is such that

M' := (M/D)[E' is a matroid of rank n - p(D) on E'. Let ~,~ff' = ~M" denote the set of hyperplanes in M'. Then the homomorphism ~,: ~:~M~ ~ T~M given by

¢(~):= ~ ,

~k(Xn,a):= a(H U O)M(a ) for n ~ , ~ ' , a ~ E ' \ n

induces a homomorphism o~: T~: ~ q~M. In particular, we have

(1.2a) ~(e~,) = e~,

(1.2b) c~(nM,(a)) = a(nUO)M(a ) for n~Y,~: ', a ~ E ' \ n .

164 A N D R E A S W . M . D R E S S A N D W A L T E R W E N Z E L

Furthermore, we have ~(~(~!) ~_ " ~ ) and a(TM, ) ~_ -~M.

Proof. For finite E this result is part of [6, Prop. 4.1]. The proof works also for infinite E; however, for the convenience of the reader we shall recall the straightforward argument.

We have to show ~b(~(~;) = {1}. Clearly, we have i~(e 2) : 8 2 = 1. Now assume E is a hyperline in M' and at, a2, a3 ~ E ' \ E are such that the

three hyperplanes H' i := a'(L' w {ai}), 1 ~< i ~< 3, are pairwise distinct, where a' denotes the closure operator of M'. Put Hi := a(H~ w D) for 1 ~< i ~< 3. Then we get

~,(e "XH, o~ " X ; ~ "XH, a~ "Xh~a, "X'~'~,a, " - t

= eM" Hi(a2)" Ht(a3)- i. H2(a~). Ha(a0- I. H3(a0. Ha(a2)- i

= I ,

because L := a(E ~o D) is a hyperline in M and

H i c ~ H ~ = L for 1 ~<i<j~<3. []

COROLLARY. Assume M' is a minor o f the matroid M. Then

(i) eM, = 1 implies eM = 1. (ii) e M, ~ ~-2, implies e~t ~ T 2 := { T 2 1 T~ TM}. []

In the sequel we recall some results of [6] and [15], which have been stated and proved only for finite E. However, all of them hold also for infinite E, because

T ~ = lim ~-~, T M = liin ~-M', M" : M I E ' M'=M[E"

T(o) lim ~-(o) M ~ UM" ===+ M'=M[E"

where E' runs through all finite subsets E' _ E of rank n and for E' ~_ E" ~_ E,

M ' = M]E', M " = M]E" the homoinorphisms TM~ ~ TM~, "]]-M'--~-[I-M" and T(~! --) T(~!, are the induced homomorphislns, defined in Proposition 1.3.

PROPOSITION 1.4. Assume eo: T~M=-) Z ar x Z E is the homomorphism, de-

fined in Definition 1.2. Assume Z ~_ E is such that every connected component of M contains exactly one e ~ Z and put E' :-- E \ Z . Then we have

(1.3) ~(Tu a~) _-_ Z E' x Z a~,

and the following sequence of abelian groups is exact:

(1.4) 0 c~ ~MT(°) c~ ~-~--% tl)(Tu ~) --~ 0.

ON C O M B I N A T O R I A L A N D P R O J E C T I V E G E O M E T R Y 165

Furthermore, we have

(1.5a) ~-U ----- -fi-~) X Z ~',

(1.6a) -D-~ ~ TM x Z ~ T~ ) x 7/~' x Z ~e.

I f in particular, the cardinality k := # E of E is finite, then we get with z := # Z and h := # ~ :

(1.5b) T M ~ "1~(0 M) X Z k-z ,

-fi-(O) X 7/k-z+h (1.6b) T ~ ~ T u x Z h ~ "M

Proof. This summarizes Theorems 1.4 and 1.5 in [6]. []

Next we shall show that for every matroid M which is representable over a field or, more generally, over a skewfield K there exists an associated homomorphism from T ~ into the abelianized multiplicative group

(1.7) Kc := K*/[K*, K*]

of K. For each x ~ K* let ~ := x. [K*, K*] denote the corresponding element in Kc. As is well known, a representation of M over K is given by a map f from E into K", considered as a left vector space over K (cf. [11, Ch. I, §4]), such that

(B) {el . . . . , e,} is a base of M if and only if {f(el) . . . . ,f(e.)} is a base in K",

or, equivalently (cf. [13, Th. 5.1.1]), by a family of maps fH: E ~ K (H~ Jet °) such that

(HI) f~l({0}) = H for all He~/t ~,

(H2) /f Hi, H2, H 3 ff J/t ~ are pairwise distinct and L := H1 n H 2 c~ H 3 is a hyperline in M, then there exist cl, c2, c3 e K * such that for all a e E we have

fnl(a) 'c l + fn2(a)'c2 + fn~(a)'c 3 = O.

Indeed, if f: E ~ K" satisfies (B) and H is a hyperplane in E, then f (H) generates a hyperplane ( f ( H ) ) in K". So there exists a left linear form @ H : K " ~ K , unique up to right scalar multiples, such that ( f ( H ) ) =

qb~l({0}). Choosing one such @n for each H ~ the resulting family fn = On o f (H ~ ~,~) satisfies (H1) and (H2). For commutative K the converse is the dual statement of Tutte's famous representation theorem [13, Th. 5.1.1] and it follows in the non-commutative case and for infinite E in the same way it has been proved for finite E and commutative K by Tutte.

Using these facts we prove (cf. [6, Prop. 3.1]):

166 A N D R E A S W . M. D R E S S A N D W A L T E R W E N Z E L

PROPOSITION 1.5. Assume M is representable over the skewfield K and

(fH)nz~e denotes a system of hyperplane functions for M and K as described above. Then the homomorphism ~k: F~--+ Kc given by

(1.7a) ¢(e) := - 1,

(1.7b) $(Xn,,,) := fn(a) for H ~ ~,~, a ~ E \ H

induces a homomorphism ¢p = (b " T ~ ~ K c. Proof. Clearly, we have ¢(e 2) = 1. Now assume that L is a hyperline in M and that al, a2, a3 ~ E \ L are such

that the three hyperplanes Hi := a u ( L u {a~}), 1 ~< i ~< 3, are pairwise distinct. We have to show that

. - 1 . . - 1 . " X - I " X t 1 3 , a , ) = i . ~,(e XH3,a2 Xn,,a2 Xn,,a~ Xr~2,~ u . . . .

Assume ct, c2, c 3 ~ K * are as in (H2). Then for i mod 3 we have

fH,_, (ai + 1)- 1 " fH, (ai +,) = -- c,_ 1" Ci 1

because of fn,+~(ai+ 1) = O. Thus we get

~1(t3" (XH13,a2 " XHt ,ax ) " - 1 . . - 1 ( S i l l , a 3 XH2,a3) (XH2,a , "XH3,a , ) )

= C 3 " c l 1 "C t ' c 2 1 " C 2 " C 3 1 = ' 1 . [ ]

REMARK. Actually, this argument suggests that it could be worthwhile to study the obvious non-commutative version of the Tutte group as well.

COROLLARY. Assume M is representable over the skewfield K.

(i) I f - 1 # 1 in K o then eM # 1.

(ii) I f -- 1 ¢ K 2, then eM ¢ T 2 . []

For binary matroids we have, with the notations of Proposition 1.4,

THEOREM 1.6. I f M is binary, then

(1.8a) ~M-r(°) = (en) _ "~{0} / t.£/27/

T u - < ~ M ) x Z e ' , (1.9a)

(1.10a)

i f the Fano-matroid or its dual

is a minor o f M

otherwise,

l r ~ ~- <~M> x z E' x z .'~.

ON C O M B I N A T O R I A L AND PROJECTIV E GEOMETRY 167

If, in particular, k = # E < ~ , then we have

(1.9b) YM ~- (eM) X ~_k-z,

(1.10b) ~ - ~ (eu~ x 7/k+h-Z

where h = # J/~ and z denotes the number o f connected components o f M.

Proof. This is Theorem 5.2 in [15] and follows from Tutte's homotopy theory for matroids as developed in [13, §6]. []

COROLLARY (Tutte). For a binary matroid M the following statements are

equivalent:

(i) M is regular, (ii) eM ~ T 2 ,

(iii) eM V ~ 1,

(iv) M contains neither the Fano-matroid nor its dual as a minor.

Proof. This is Theorem 4.2 in [15] but it follows also directly from Theorem 1.6. (See, for instance, Corollary 1 of Theorem 3.1 in [6].) []

REMARK. For obvious reasons, we consider this corollary as a 'Tutte group theoretical' interpretation of Tutte's characterization of regular matroids, stated in [13, Th. 7.4.1].

2. P R O J E C T I V E GEOMETRY FOR MATROIDS

In this section we want to show that cross ratios, as defined for certain configurations of points, lines or other subspaces in a projective space over a skewfield (see for instance [2, Ch. III, §4] or [12, §4.2]), have their natural habitat in the context of matroid theory, taking their values in the inner Tutte group, defined above.

We continue to assume that M is a matroid defined on E of finite rank

n ~> 2 with ~ = ~ M its set of hyperplanes, p = PM its rank function, and a = aM its closure operator. Our starting point is

LEMMA 2.1. Assume H1, H 2 6 J t °, p(H1 ~H2)= n - 2, a, b ~ E \ ( H 1 uH2) and p((H 1 c~ H2) w {a, b}) = n - 1. Then we have

(2.1) Hl(a) .H2(a )-1 = Hl(b )" H2(b )- 1.

168

Proof. have

ANDREAS W. M. DRESS AND WALTER WENZEL

For H a := tr((H1 n H2) u {a, b}), a i e H l k H 2 and a2 e H 2 k H 1 we

Ha(a)" H2(a)- l

=eM. Hl(a2). Ha(a2)- 1 . Ha(al). H2(al ) - 1

= Hl(b ) . H2(b ) - 1.

(Cf. Lemma 1.3 in [6]).

Lemma 2.1 justifies the following definitions.

[]

D E F I N I T I O N 2.2. Assume Ha, H2, H a are hyperplanes in

p(H 1 n H 2 n Ha) = n - 2 and H 3 ~ Hi for i e {1, 2}. Then we put

(2.2) H i H3 H2 := nl(aa) .H2(a3 )_ l

where a 3 E H 3 \ ( H 1 u H2).

M with

If Hi , H2, Ha are as in Definition 2.2, then we have

H3 Ha = 1 (2.3) H1

and

Ha H2 = t H 2 H l - a (2.4) Ha [ H3

Furthermore, for H1, H2, H3, H e ~ with H ~ H i for 1 ~< i ~< 3 and p(H1 n H 2 n H 3 c~ H) = n - 2 we have

(2.5) Ha H H2 . H2 H Ha " Ha H H1 = 1.

Finally, for pairwise distinct Hi , H2, H3 e #g with p(H~ n H 2 n Ha) -- n - 2 the very definition of T ~ implies

(2.6) H1 . H3 H3 H 2 . H2 Hi H3 H2

D E F I N I T I O N 2.3. Assume H1, H2, H3, H4 with p(H 1 n H 2 n H 3 n H4) = n - 2 and Hi ,

{/-/a, /-/2} n {H3, H4} = ~Z~. Then the cross ratio I-I3

H1 = ~ M .

are hyperplanes in M

H 2 ~ H 3 , H4; that is

s-g~) is defined H4

ON COMBINATORIAL AND PROJECTIVE GEOMETRY 169

by

HE

I f i l l , H 2, Ha, H4 are as in Definition 2.3, then we have for a 3 6 H3\(H 1 U H2)

and a4 e H4\(H1 u Hz):

(2.7a) [ H 1 H 2 ] = H l ( a a ) ' H 2 ( a a ) - l " H 2 ( a ' ) ' H l ( a 4 ) - l ' H a H4

The following trivial lemma lists a few simple rules concerning these cross ratios.

LEMMA 2.4. Assume L is a hyperpline in M and H1, H2, Ha, H4, H5 e contain L and satisfy {H1, H2} c~ {Ha, H,~} = ~ .

O) We have always

H3 H4A H4 Ha

H21 = 1. I-I4

If H I ---- H 2 or H a = H4, then [[H1 (ii) Ha

I ._

(iii) I f H 5 q~ {HI, H2}, then

(2.9a, [H: H : ] . [ H : H : ] = E H :

(iv) xf ns¢{na, n4}, then

Furthermore, the following identities hold:

PROPOSITION 2.5. Assume L is a hyperline in M and H~, H2, Ha, H4 e Jf contain L such that {n~, n~} ~ {n~, n~} --- ~ .

(i) We have

H4

170 ANDREAS W. M. DRESS AND WALTER WENZEL

(ii) I f H1, H2, H3, H A are pairwise distinct, then we have

(2.11a)

(2.11b)

Proof (i)

[# -a -a L]=,,,, n,J'Ln, H,J'LH~

[~: H21 FH1 Hal,[H2 ~n:] = a M.

By (2.8) it suffices to prove (2.10a). Assume a ~ H i \ L for 1 ~< i ~< 4. By Lemma 2.4(ii) we may assume that H2, H2, Ha, H a are pairwise distinct. Then we get by (2.7a) and the definition of " ~ :

I H 1 H z ] = ( H l ( a a ) ' H 2 ( a 4 ) - 2 ) ' ( H 2 ( a 4 ) ' H 2 ( a a ) - 2 ) H a H4

= (eM" H4(aa)" H4(a2)- 2. Ha(a1). Ha(a4)- l)

× (aM" na(a4) 'na(a2)- 2 • n~(a2) • n4(aa)- 2)

= Ha(al) .n4(a D- 1. n4(a2)" Ha(a2)- 2

=[.': 8 (ii) By (2.10b) it suffices to prove (2.11a) which follows at once from (2.7),

(2.6) and (2.5):

[L H~_I LH1 H4_I'LH~ =/H1/ Ha H2.H2 Hi

×(. H2 H4 Hi. Ha H4

=8 M" 1

a M . []

Hal. Ha H2 H i )

H2 .[Hi H4 Ha)

The following results demonstrate the significance of cross ratios in the context of representability theory of matroids.

PROPOSITION 2.6. Assume M is representable over some (commutative) field K. Let ( fn )n~e denote a system of hyperplane functions for M and K and assume tp: qF~--* K* is the homomorphism induced by Proposition 1.5; this means we have

(2.12a) ~o(eu) = - 1,


(2.12b) ~o(H(e)) =fn(e) for H ~ , eeE\H.

Furthermore, assume that H t, H 2, Ha, H4 are hyperplanes in M with p(H 1 ni lE n H a r~H4) = n- - 2 and {Ht, H2} r~ {Ha, Ha} = ~ . Then the following three statements are equivalent:

(i) q~ Ha H4

( i i ) [ H ~ H 2 ] Ha Ha ~ 1,

(iii) H1 ~ H2 and H 3 ~ H 4.

In particular, M cannot be representable over any field, if there exist four pairwise distinct hyperplanes HI, H2, H3, H4 ~ J f with

p ( H l c ~ H 2 ~ H 3 n H 4 ) = n - 2 and E HtHa H4H2] =1"

Proof This is a reformulation of [6, Prop. 3.2], but - reformulated in this way - it is also a simple consequence of well-known facts from projective geometry. []

THEOREM 2.7. (i) .u-r(°) is the subgroup of TiM generated by ~ and all cross

[ H1 H21forwhich H1, H2, Ha, H4arepairwise distinct hyperplanesin ratios H3 Ha

M with p(H 1 n H E c~ H 3 c~ Ha) = n -- 2.

(ii) -r(o) is generated by all cross ratios with H1, H2, Ha, H4 as in uM H4

(i) if and only if M is not regular. Proof (i) is a reformulation of the fundamental Proposition 2.90) in [15]

which translates the more elementary part of Tutte's homotopy theory into the language of Tutte groups.

(ii) For binary matroids the asserted equivalence follows from Lemma 2.4(ii) and Theorem 1.6 together with its corollary, because there do not exist four pairwise distinct hyperplanes which intersect in a hyperline. If, however, M is not binary, then there exists some hyperline which is contained in four pairwise distinct hyperplanes, and then Proposition 2.5(ii) shows that eM lies in the group generated by all cross ratios. Thus in this case we are done by (i). []

Theorem 2.7 suggests a more thorough examination of cross ratios in Tutte groups from the point of view of projective geometry. In the sequel let &a = ZfM denote the set of hyperlines in M.


DEFINITION 2.8. Put

(2.13) ~vft4): = {(HI, H2, Ha, HA)egAIHI AH2 nHa ni l^

= H ~ n H j e ~ fo r ie{1 ,2}and je{3 ,4}} .

(i) (H1, H2, Ha, Ha) , (H~, Hi, Hi, H~) 6 .~,(4) are called projective neighbours (hA H' of each other, if n4= 1 Hi ~ i l i= 1 i and there exists some H e ~f~ with

(I) n~=l H i ~ H, n~=~ H'i ~ H and (II) for 1 ~< i ~< 4 we have Hi = H~ or Hi n H; = Hi n H = H~ n H e L,e.

In this case we write (H1, HE, H3, HA)^(H'I, H~, H~, H~) or, more H

specifically, (Hi, H2, Ha, HA) A (H'~, Hi, Hi, H~) or, still more specifically, H

(H1, H2, Ha, HA) A (H'~, Hi, Hi, H~), if L := n4=l H~ and L' := n4=~ H'i. LL '

(ii) (H1, H2, Ha, HA), (H~, Hi, Hi, H~)~A~t'0 are called projectively if there exist k t> 0 and (Hli, H21 , H3~, H4i)e~f ~tA) for 0 ~< i ~< k equivalent,

such that

and

(Hlo, H2o, Hao, HAo) = (H1, H2, H3, HA),

(Hlk, H2k, H3k, HAk) = (H'x, H'2, H'3, H'4)

(H1 i-1, H2i-1, H3i-1, Hgi-1) ^ (Hli, H21, H31, HAl)

for l~<i~<k.

In this case we write (H1, H2, Ha, H,I. ) pr (H~, Hi, Hi, H~).

REMARK. If(H1, H2, Ha, H4) ^ (H~, Hi, Hi, H~) and Hi = H~, H i = H) for some i, j with 1 ~< i < j ~< 4, then we have H i = Hj = H'i = Hi, because there exists at most one hyperplane which contains the two distinct hyperlines

L : = N ;1 xi and L' : - n~,=l H'i.

Figures 1 and 2 demonstrate projective neighbours in the case n = 3. Next we prove the following simple, but fundamental

THEOREM 2.9. Assume (H 1, H2, H3, H4) ~ (H'I, H'2, H'3, H'4). Then

HA x J" Proof. Clearly, we may assume (H I, H2, H3, HA) ^ (H'I, Hi, H~, H~). Put

4 A L:=NHi, L':=NH;

i=1 i=l

and assume H e ~ is as in Definition 2.8(i).


~ 1 4 "

a4 Fig. 1.

.& '~ =~

a4

Fig. 2.

If Hi = H~ for some i t { l , 2} and Hj = H~ for some j~{3, 4}, then the remark after Definition 2.8 would imply at once (H1, Hz, H3, H4)q~ 3~ (~) and (H'~, H'2, H'3, H'4)¢ ;,~(4~ Thus by Proposition 2.5(i) we may assume H3 ~ H~ and H4 ~ Hi.

For j ~ {3, 4} choose some aj e (Hi n H)\L; this is possible, because (I) and (II) imply that Hj c~ H and L are distinct hyperlines in M. (See Figure 1 and Figure 2 in the case n = 3.)

We shall now show that

(2.14a) ni(a3).H~(a~.) -1 = a'i(as)'n~(a4) -1 for i t { l , 2}.

This is obvious in the case H~ = H}. Otherwise we have

= ~((H, c~ H',) ~ {aj}) for j e {3, 4},

174 A N D R E A S W. M. DRESS A N D W A L T E R W E N Z E L

g~ G g2 g3

Fig. 3.

and then (2.14a) follows from Lemma 2.1. Applying (2.14a) twice we get (2.14). []

As suggested by classical projective geometry (cf. [11, Th. 2.6]), Theorem 2.9 can be used to derive the following Tutte group version of the Pappos theorem:

T H E O R E M 2.10 (see Figure 3). Let M denote a matroid of rank n >1 3 defined on E. Assume furthermore that h 1, hz, h3, 91, 02, 93, aa, a2, a3 are pairwise distinct hyperlines in M such that the followin9 fourteen subspaces are hyperplanes in M:

(2.15a) H : = a(h 1 u h 2 u h3),

(2.15b) G:= a(g lug2ug3) ,

(2.15c) Ais:= a(h iug juak) for {i,j, k} = {1, 2, 3},

(2.15d) K~:= tr(a 3 u a i ) for ie{1, 2},

(2.15e) Gi:=tr(alwgi) for 1~< i<~ 3,

(2.15f) L := tr(a 3 u h3).

Assume, furthermore, that gi q~ H and that hl ~ G for i = 1, 2, 3. Then we have

( 2 . 1 6 ) = 1. LK1 K 2 J


Proof First we remark that our assumptions imply ai ~ H and ai ~ G for i = 1, 2, 3. To prove our claim it is enough to show that

[A z A:,] FA z A I] (2.16a) LK, G A=LK

But

G (hi 2, a21, g l , G3) /~ (A32, Gx, K1, A23 )

a3 al

and

A21

(A32, G 1, K1, A23)a[~h3(A32, A31 , L, H)

and, similarly, exchanging 1 and 2,

and

G (A21'/112' K2' G3) A (A31, G2, g2, A13)

a3 a2

AI2

(A31, G2, K2, A 1 3 ) a2/~h3 (A31, A32 , L, H).

Hence, Theorem 2.9 implies

=FA21 Ax21-1 FA12 A211, LK: =LKz

as claimed. []

REMARK. Theorem 2.10 together with Proposition 2.6 implies the Pappos theorem; that is, it implies that the non-Pappos matroid is not representable over any field.

In Definition 2.3 we have defined the cross ratio F H1 Hzl for H1, H2, n3, k H3 H4/ H 4 e ~ with p(H1 ~ H2 n H3 n H4) = n - 2 and {H1, H2} n {n3, n4} = ~ .

We shall n°w sh°w that much m°re generally cr°ss rati°s IPaQ1 Q2P21 =-n-t°)~°M

exist for all appropriately defined modular quadruples (PI, P2, Q1, Q2) of subspaces of M. First we recall the following

176 A N D R E A S W . M. DRESS A N D W A L T E R W E N Z E L

DEFINITION 2.11. Two subspaces P, Q of M constitute a modular pair (P, Q) in M if

(2.17) p(P u Q) + p(P c~ Q) = p(P) + p(Q).

DEFINITION 2.12. Assume 0 ~< l, m ~< n. A quadruple (P1, P2, Qx, Q2) of subspaces of M is called a modular quadruple of type (l, m), if

O) P(P1) = P(P2) = l, p(Qx) = p(Q2) = m

and for i, j ~ (1, 2} the following assertions hold:

(ii) (Pi, Q~) is a modular pair, (iii) Pic~Q~ = P1 c~P2nQ1 c~Q2, (iv) ~7(P i u Qj) = tr(P 1 u P2 u Q1 u Q2).

cross ratios [P1 P2] for modular quadruples (PI, P2, QJ, Q2) To define o f Q~ Q~ ._1

subspaces of M we shall need the group -U-~ defined in [6, Def. 1.2]. This group will be used later on once more. We recall the definition:

D E F I N I T I O N 2.13. Let ~:~u denote the free abelian group generated by the symbols e and Xt~ ~ ....... ) for (al . . . . . a,} ~ and let K~ denote the subgroup of ~:~ generated by e2 and all elements of the form

• X -~ for {al . . . . a,} ~ , 8" X ( a I . . . . . an) (a~t~,.,.,a~.))

z an odd permutation in Z.,

and

S - 1 • - 1 . X(at,...,an_2,b2,c2). (al,...,a,_2,b2,cl) X(at, . . . ,an_ 2,bl,cl) X(al,...,an-2,bl,c2)

if (ax, • • •, a._ 2, bi, c j} E ~ for i ,j e {1, 2}, but (ax . . . . . a._ 2, b~, b2} ~ ~. Then T~ is defined by

~-~ := U=M/~M.

CONVENTION• Let v: U=~--~ T~ denote the canonical epimorphism and

put

(2.18a) e~ := v(e);

(2.18b) T~a ........ ):= v(X~ ........ )) for {a t , . . . , a,} e ~ .

The following result compares the groups T u and T~.

O N C O M B I N A T O R I A L A N D P R O J E C T I V E G E O M E T R Y 177

PROPOSITION 2.14. We have a well-defined monomorphism co: TM ~ 7~M given by

o~(~,):= ~, o ) ( n ( a ) " n ( b ) - 1 ) : = T( a ...... a . _ . a ) " - 1 T ( a . . . . , a. _ ~,b)

for ne,,vg, a, beEkH and a({al . . . . . a~-l}) = H.

denotes the obviously well-defined epimorphism If, furthermore, fl: T~ --~ 7/ given by

p ( ~ ) : = o,

fl(T(~ ....... )) := 1 for {al , . . . , a,) e ~ ,

then the following sequence of abelian groups is exact:

o~ v ~ ~ z ~ o .

In particular, we have

Y~M ~T= M x Z.

Proof This result summarizes Theorems 1.1 and 1.2 in [6] (there these results have been stated for finite E, only; however the proofs work also for infinite E). []

In the sequel we identify each ]re 7M with ~o(T) e 7~ . In particular, from now on e~ --- ~, . Thus we consider 7M as a subgroup of 7~ .

REMARKS• (i) If D is a subspace of M of rank m and {ax . . . . . am}, (bl . . . . . b,,} are bases of the restriction MID, then for all era+ 1 . . . . . en, f,,+ 1, . - . , f . EE with

a(D w {em+l . . . . . e.}) = a(D w {fm+ 1 , ' " , f.}) = E

we have

T - 1 T(al , . . . ,a , ,e~+ l , . . . ,e ,) " (al , . . . ,a , , fm+ l . . . . . f . )

T(bt , . . . ,bm,em+l, . . . ,e ,) - 1 • T ( b l , . . . , b . , f " . . . . . . . f . ) "

This follows from Definition 2.13 together with a repeated application of the base exchange axiom o r - a s wel l - f rom Proposition 2.14.

(ii) Let 0: T~M--* 7/E denote the obviously well-defined homomorphism given by

O(~u) := O,

O(T~ .. . . . . . . . )):= ~ 5,, for {a, . . . . . a.)e~M i = 1

where


where, of course, for e e E we denote by fie ~ Z/~ the map given by

{10 for f = e ~e(f) := f o r f ¢ e.

The results of [6, §1] imply that -u-r(°) = ker 0.

To generalize our definition of cross ratios we introduce one further notation. We put

(2.19) ~(m := {(al, • • ", a,) e E" [ {at . . . . , a,} e N};

thus an element a ~ ~(u) is an ordered base of M.

D E F I N I T I O N 2.15. Assume 0 ~< l, m ~ n, and let (Pt, Pz, Qt, Q2) denote a modular quadruple of subspaces of M of type (l, m). Then the oeneralized

[P1 P2]~T(°) is defined by cross ratio Q1 Q2I u

and

(el . . . . . e0 is an ordered base of M[(Ps n P2 n Q1 n Q2),

(ft, . . . , f ,) is an ordered base of M/(P1 u P2 u Qs u Q2)

Pn, . . . , PlsE Pi, qjl, . . . , q jt ~ Qj

are such that

aii := (el . . . . . ev, Pn . . . . , Pi,, q j t , . . . , qjt, f l , . . . , f , )e~(M~

for i, j e { 1 , 2}.

REMARKS. With the notations of Definition 2.15 we have clearly v + s = l,

[P1 P 2 ] i s w e l l d e f i n e d b y R e m a r k ( i ) v + t = m a n d v + s + t + # = n . Q1 Q2

following Proposition 2.14 applied to, say, D = Pt and D = P2- Furthermore,

IP1 P2-] T(o) the subsequent Remark (ii) shows that Q~ Q2] ~ u .

EXAMPLES. (i) By definition we have

Q1 Q2 = 1 if P1 = P 2 o r Q t = Q v


In particular,

[P1 P 2 ] = l Q~ Q~ if P1 = P2 = E or Qi = Q2 = E

or P1 = P2 -= a (~ ) or Q1 = Q2 = a(~2~).

(ii) Assume (Hi, H2, H3, Ha)eJ / f (4~. Then l-/Hx H21-1 as defined in L H3 H , d

Definition 2.3 on the one hand and in Definition 2.15 on the other hand coincide, because for L : = H 1 c~H 2 c~H 3 n i l , , any base {el . . . . . e,-2} of MIL, and a~eHi\L for ! ~< i ~< 4 we have by Proposition 2.14

Hi (a3 ) . Hi(a4)- i . H2(a4)" n2(a3)- i

= T(( e ...... e._2,al,a3)'Z(ell,. . . , e ._2,al ,a,) 'Z(e ...... e._ 2,a2,aa)'T(e~... , e.-2,a2,a3)"

Definitions 2.13 and 2.15 imply immediately that for any modular quadruple (P1, P2, Qi, Q2) we have

(2.21a)

and

Qi Q2 Qi Q2J Q2 QLI Q2 Qi

P2 Q2

To study generalized cross ratios more thoroughly we state

LEMMA 2.16. Assume 0 <<. l, m <~ n and (P1, P2, Qi, Q2) is a modular quadruple of subspaces of type (l, m).

(i) I f m < n, eoe(e l c~ Pg\(Qa u Q2) and Q'i:= a (Q,u {e0})for i~{1,2}, then (PI, P2, Q'l, Q'2) is a modular quadruple of type (l, m + 1), and we have

Q~ O~ Qi Q~I"

(ii) I f l < n, eo~(Q1 c~ Q2)\(P1 u P2) and P',:= a(P,u {eo} ) for ie{1, 2}, then (P'I, P'2, Q1, Q2) is a modular quadruple of type (l + 1, m), and we have

(iii) I f l, m < n, eo e e \a (P i w P2 t..) Ol u O2) and P'i:= a(Pi u {eo}),

180 A N D R E A S W . M. DR E S S A N D W A L T E R W E N Z E L

Q'~ := tr(Q~ u {eo}) for i e {1, 2}, then (P'~, P'2, Q'~, Q'2) is a modular quadruple of type (l + 1, m + 1), and we have

LQ, ~i] Proof. All assertions follow trivially from Definitions 2.12 and 2.15. []

CONVENTION. If a, b e E are no loops in M and D~, D2 are subspaces of M such that (D , D2, a({a}), a({b})) is a modular quadruple, then we put

and, of course, similarly

a [o, L]--r L DI D2 J"

Moreover, if al, a2, a3, a , e E are such that (a({a,}), a({a2}), a({a3}), tr({a,}))is a modular quadruple, we put

at a2 . [a~ a,] "= [~((a'D L~({a3}t

EXAMPLES. (i) Assume a, b e E\(H 1 u H 2 ) , put

~ := ~(L ~ {a})

~({a,})J"

H~, H 2 e ~ with L := H 1 c~ H 2 ~ ~ and

and H 4:= a(L u {b}).

Then a repeated application of Lemma 2.160) implies

(ii) Assume al, a2, a3, a4eE satisfy a({ax, a2, a3, a4}) = tr({a,,aj}) = 2 for ie { 1, 2} and j e {3, 4}. Choose any hyperline L in M with a(L u {al, a2, a3, a,}) = n and put Hi :-- a(L u {ai}) for 1 <~ i <~ 4. Then a repeated application of Lemma 2.16(iii) yields

DEFINITION 2.17. (i) If (P1, P2, Q1, Q2) and (P1, P 2 , Q'I, Q2) are as in Lemma 2.16(i), then (P1, P2, Q1, Q2) is called a specialization of (Px, P2, Q'I, Q'2) of the first kind.


(ii) If (P1, Pz, Q1, Q2) and (P], P~, Q1, Q1) are as in Lemma 2.16(ii), then (P1, P2, Q1, Q2) is called a specialization of (P'l, P'2, Q1, Q2) of the second kind.

(iii) If (P1, P2, Q1, Q2) and (P], P~, Q], Q~) are as in Lemma 2.16(iii), then (P1, P2, Q1, Q2) is called a specialization of (P'I, P'2, Q'I, Q'2) of the third kind.

(iv) Two modular quadruples (PI, P2, Q1, Qz) and (P~, P'2, Q], Q[) are called equivalent, if there exist k ~> 0 and modular quadruples (Pli, P21, Qxi, Q2i) for 0 ~< i ~< k such that

(Plo, P20, Qlo, Q20) = (P1, P2, Q1, Q2),

(PIR, P2k, Qlk, OEk) = (Pi, P'2, Ol, 0'2)

and for each i with 1 ~< i~< k the quadruple (P1 i-1, P2 i-1, Q1 i-1, Q2i-1) is some specialization of (Ply, P2i, Qll, Q2i) or vice versa. In this case we write

(P1, P2, Q1, Q2) ~ (PI, P~, QI, Q~).

Lemma 2.16 implies

P R O P O S I T I O N 2.18. Assume (P1, P2, Q1, Q2) "~ (P'I, P'2, Q'I, 0'2). Then

Q1 Q2 LQ'I QiJ"

As a special case of Proposition 2.18 we get

P R O P O S I T I O N 2.19 (see Figure 4). Assume a~, a2, a3, a~, a'l, a2; as; a'4 e E are such that

:= a2, a3, a,}) and G2 := a'2,

are subspaces of rank 2 and p({ai, aj} ) = p({a'i,a~})= 2 for ie{1,2} and j • {3, 4}. Furthermore, assume that there exists some s • E\(GI u G2) such that p({s, ai, a~}) = 2for 1 <. i <. 4. Then

[a, a2]=[a'l a'21 a3 a4 La'3 a'4J"

B1

82

~ , a 3

Fig. 4.

,a'~

..a~

a~

182 A N D R E A S W. M. DRESS AND W A L T E R W E N Z E L

L22

el

L,~I = L12

Fig. 5.

Proof Our assumptions imply n t> 3, because otherwise there does not exist any s satisfying our conditions. Choose some hyperline L in M with s e L and p(LwG1)=n . Then we have also p ( L w G 2 ) = n , because G1 - a({s} w G2). Put H, := a(L w {a,}) for 1 ~< i ~< 4. Then we have also a'i ~ Hi and thus Hi = a(L w {a~}) for 1 ~< i ~< 4. Now Example (ii) following Lemma 2.16 implies

[al a21 IH1 H21=Fa'I d21 a3 a, = H3 H 4 [_a'a a'4_]" []

The last results suggest a close connection between cross ratios in Tutte groups and cross ratios of four pairwise distinct collinear points in a projective space over a skewfield as defined in [2] or [12]. This will be discussed in the next section when we have studied the Tutte group of such spaces. However, at the end of this section we want to state one more result which fits nicely into our set up.

PROPOSITION 2.20 (see Figure 5). Let M denote a matroid of rank n >I 3 defined on E. Assume that K1, K2, G1, G2 are pairwise distinct hyperplanes in M and e i e Gi\(K1 u K2) for i e { 1, 2}. Furthermore, assume that Lis := Ki c~ G s is a hyperline in M for i, j e {1, 2} and K1 n G1 = K1 n G 2 = G 1 N G 2. Finally, assume that there exists some ea e KI\L1 ~ with

H1 := a((G2 n K2) w {el}) = a((G2 c~ K2) u {ea})e g u ,

H2 := a((G1 (~ K2) u {e2} ) = (r((G 1 c~ K2) u {ea})~ ~M.

Then we have

(2.23) Kl(e O. Kl(ez)- 1. Kz(ez). K2 (e~)- 1 = ~M.

In particular, if L = K1 n K2 is a hyperline and if F i := a(L u {el}) for i = 1, 2, K2

then F1 F2 =eM.

Proof. By Lemma 2.1 we have

Gl(ea). G l ( e 2 ) - 1 = K 2 ( e a ) . K 2 ( e 2 ) - 1


and

G2(el). G2(e3)- i = K2(el).K2(ea)- 1.

Thus we obtain

(K l(e l) . K l(ez)- 1) . ( K 2(e~) . K2(e l)- i)

= eM. G 2 ( e l ) . G2(e3) - 1. Gl(e3). Gl(e2)- 1

× K2(e2 ) " K2(ei)- 1

=(eM' G2(el) • G2(e3)- 1)" (K2(e3)" K2(ei)- i)

= ~M" [ ]

This result reflects the well-known facts about harmonic quadruples of collinear points in a projective plane in the much more general framework of Tutte-group theory for combinatorial geometries.

3. T H E T U T T E G R O U P OF P R O J E C T I V E S P A C E S

In the sequel we assume that ~ is a projective space with E as its set of points and ~# as its set of lines. Thus by the definition of a projective space the following axioms hold:

(P1) Any two distinct points x, y ~ E are on exactly one line xy ~ ~.

(P2) I f x, y, z, w are four distinct points, no three on a common line, and if xy intersects zw, then xz intersects yw.

(P3) Each line contains at least three points.

The following elementary result relates projective spaces and simple matroids, where, as usual, a matroid is called simple if and only if every circuit has cardinality at least 3.

P R O P O S I T I O N 3.1. I f M is a simple matroid on E with the followino properties:

(PI') #(G)/> 3 for every flat G of rank 2 in M, (P2') G n H # ~ for every flat G of rank 2 and every hyperplane H in M,

then E together with the flats of rank 2 as set of lines and the obvious incidence relation forms a projective space, and every projective space can be derived from a simple matroid satisfyin9 (PI') and (P2') in this way. Moreover, any pair (P, Q) of subspaees of M is necessarily modular in this case.

Proof This is (essentially) Theorem 3 in [14, §12.1]. []


From now on we assume that ~ has dimension n - 1 ~> 2; this means that the

matroid M = M(~) induced by ~ has rank n t> 3. As above, let W = WM

denote the set of hyperplanes, ~ = ~ M the set ofhyperlines, p = PM the rank function and o- = aM the closure operator of M.

In the sequel we want to study the Tutte group of the projective space M.

To this end we shall make use of perspectivities and projectivities as defined in classical projective geometry. For any hyperline L E ~ we put

(3.1) ~ L : = {H~ , ,~ IL~_H} .

D E F I N I T I O N 3.2. Assume L1, L 2 ~ ~ and H o e ~ contain some subspace D of corank 3 and Ho contains neither L1 nor L2. Then the bijection

P = P~LI,no,L2): Of L~ --* ~L~ defined by

(3.2) p(H) := a((n n no) w L2)

is called a perspectivity from alL1 onto O~L2. Any composition of perspectivities in M = M(~) is called a projectivity.

REMARK. I f L a = L2 in Definition 3.2, then P~L~,no,L~) is clearly the identity

map defined on Jt°Lc

NOW we have the fundamental

P R O P O S I T I O N 3.3. Assume LI and L 2 are hyperlines in M. Then for all pairwise distinct Ha, H2, H3 ~ agg~L~ and all pairwise distinct H'I, H'2, H'a ~ ~L~

there exists some projectivity p: ~ L I -~ ~'~L2 with p(Hi) = H'i for 1 <<. i <~ 3. Proof We proceed by induction on n. For n = 3 this result is the dual statement to [-12, §1.2, Prop. 6]. If n/> 4,

then we choose some hyperline L 3 ~ ~ which intersects L~ and L2, say x i ~ L i ~ L 3 for i ~ { 1, 2}, and apply ou r induction hypothesis to the projective geometries ~ i induced by the lattices of subspaces containing x~ for i ~ { 1, 2}. (Of course, it suffices to consider the case n = 4 in our induction step, because

for n ~> 5 we have L1 c~ L 2 ~ ~ . ) []

REMARK. It is here where we need that the dimension of the projective

space ~ is at least 2.

Now we can show

L E M M A 3.4. Assume that L, U~LP and H1, H2, Ha~Jg~L are pairwise distinct and that H'I, H'2, H'3, H'4 ~ oVf z, satisfy {H'I, H~} n {H~, H~.} = ~ . Then there exists some H4E~Cf z with H 4 ~ H i f o r i6(1, 2} and

(3.3) [H, H2]=FH ~ H'2].


Proof In case H'I = H~ or H~ = H i we may put H4 := Ha. Otherwise, by Proposition 3.3 there exists some projectivity p: ;'~¢gL' ~ Jt~L with p(H'i) = Hi for 1 ~< i ~< 3. So we can invoke Theorem 2.9 to conclude that (3.3) holds for H4 := p(H'4). []

Next we prove

T H E O R E M 3.5. Assume Lm~LP, and Hi, H2E~L are distinct. Let fgtm,n2) denote the permutation 9roup of all projectivities p: ~L--~ #t°L with p(Hi)=Hi for i~{1,2}. Then for H3mZ-t~L\{Ha, H2} the map

~r(o) defined by ~ ~ ( H 1 , H 2 ; H 3 ) : ~(Ha,H2) ~ u M

(3.4) c~(p):= Ha p(H3)

is a 9roup epimorphism which does not depend on H a. In particular, for all pairwise distinct hyperplanes Ha, H2, Ha with H a ~ H 2 c~ n 3 ~ ~ we have:

(3.5) -n-(o) H1 - ~ t = H a H Ha C~ H 2 t~ H a _ H ~ ~ \ { H 1 , H2 •

Proof By (2.9a) and Theorem 2.9 we have for p, q ~ f#(H,n~):

H 2 p,q,.3,,] p(H3)J LP(H3) P(q(H3))l

=[.: .2 F., = a(p). a(q);

thus a is a homomorphism. Lemma 3.4 and Theorem 2.7 imply that a is surjective, because projective spaces of dimension at least 2 are not regular. In particular, (3.5) holds.

It remains to show that a = a(H1,H2;H3) does not depend on H3. Assume H4~3~L\{H1, H2, Ha} and put a ' := a(nl,H2;n4). By Proposition 3.3 there exists some q ~ ff(H~,H~) with q(H4) -- H 3. Now Theorem 2.9 and (2.9a) yield for

P ~ ~(//~,n2):

~(p) = a(q o p o q- 1)

H 2 H2 =[-Ha_H3 q(p(q_l(H3)))l =[q(H14) q(p(H4)) 1 = [ H : p(H~) 1

=~'(p). []

186 A N D R E A S W. M. DRESS AND W A L T E R W E N Z E L

Using Theorem 3.5 we shall now show

T H E O R E M 3.6. The inner Tutte group -~) of the matroid M = M(~), associated with the projective geometry ~ = ~ . _ I(K) of dimension n - 1(>~ 2) over a skewfield K is canonically isomorphic to the abelianized multiplicative

group K~ of K:

(3.6) "M-r(°) =" K, = K*/[K*, K*].

In particular, if K is a field, then

(3.6a) T~ ) -~ K*.

Proof. Choose some system of hyperplane functions (fn)n ~ ~r for M and K and consider the homomorphism q~: T ~ K c given by Proposition 1.5. We want to show that -,-(o) q~lT~,: u u --. Kc is an isomorphism.

For a map f: E ~ K we put kerf : = f - 1({0}) if f - 1({0}) e ~ . Choose some fixed H1, H2 e Jet ~ with H1 # H2. For x e K * we get with H3:= ker( f , , - f - 2 )

and n 4 := ker(fn~ • x - f,2):

(3.7) ~o H3 H4

because for y l e H 3 \ ( H l n H 2 ) we have fn~(yO=fu,(yO, while for Y2 eH, \ (H1 c~ H2) we get fH~(y2) =fn~(Y2)'x. Thus we have ~o(~-~ )) = Kc. It remains to show that q~ is injective.

Let ~: f~(n~,n~) -~ T~ ~ denote the epimorphism given by Theorem 3.5. The

map g: K* ~ f¢(H,,n,): X ~ g(x): a'~Hlc~H2 --+ J/t~H,c~Hz, given by

(3.8) ker(fn, .c 1 + f , , . c2)~-*ker( fn , 'x .c I + fn~.c2)

for (cl, c2)eK2\{(0, 0)}

is well defined and hence a group homomorphism. Indeed, it is standard and easily seen that g(x) is a composition of two perspectivities for every x eK*:

Put L1 := H1 n H2, choose some hyperplane H'I e ~ with L 1 ~ H'~ and put

L2 := H't n n2, no := ker(f,~ - f i l l ) ,

H~ := ker(fnx "x - fn i ) .

Then with the notations of Definition 3.2 we have

g(x) = P(Lz,H'o,Lt) o PtLt,Ho, L2)"

By (3.5) the composition ~ o g: K* ~ 1-~ ) is surjective, and together with (3.7)


we see that ~0 o a°9 : K*-- -K~ is the canonical epimorphism re. Thus the following diagram of epimorphisms commutes:

K * ~ ~' K~

Since "u-r~°) is abelian, this means that the epimorphism ~o: 1-~ ) --~ K c must be an isomorphism, because otherwise Kc would not be the unique maximal abelian factor group of K*. []

Now we are able to discuss the connection between cross ratios in Tutte groups and cross ratios of four pairwise distinct collinear points in a projective space M = M(~) over a skewfield K of dimension n - 1 with n >~ 3. Suppose that P, Q, R, S are four pairwise distinct points on some line G

in M. In [2, Ch. III, §4] the cross ratio [ P Q] ~ K~ is defined to be the class

s. [K*, K*] if and only if there exist p, q E K" satisfying

(3.9) P = K . p , Q = K . q , R = K . ( p + q), S = K . ( p + s.q).

Now assume that (3.9) holds and choose some hyperline L in M with p(L w G) = n.

Put H1 := a(L u {P}), H2 := a(L u {Q}), Ha := a(L w {S}) and H 4 := a(L u {R}). Then we can obviously choose some system of hyperplane functions (fn)n~Jr for M and K such that

fn , (R) = fn2(R) = fn2(S) = 1, in , (S) = s.

If cp: -~ ) q* K~ denotes the isomorphism given by Theorem 3.6, we obtain by Example (ii), following Lemma 2.16,

=fn,(S) "fn2(S)- 1 .fn2(R). f n , (R ) - 1. [K*, K*]

=s" [K*, K*].

Thus by identifying each T~ T~ ) with q~(T)~ K~, we see that the cross ratio i - - _ _

/.~ ~ / in the classical sense coincides with the generalized cross ratio I . . . ~

I S Q I as defined in Definiti°n 2"15"


At the end of this section we want to point out a simple, but remarkable

consequence of a result by Grundh6fer:

T H E O R E M 3.7. I f ~ is a finite non-Desarguesian projective plane, then for

M = M ( ~ ) we have

T(o) {1}. M =

Proof. Assume L is some point in M. Then by the dual statement of

[-9, Th. 2] the group of projectivities from ~ L onto ~f~L is 5-transitive and

thus in particular 4-transitive. Since the order of ~ is at least 4, our assertion

follows directly from Theorem 2.9 and (3.5). [ ]

Summarizing Theorems 3.6 and 3.7 we get

C O R O L L A R Y 3.8. For every finite projective space M = M ( ~ ) o f dimension

at least 2 the associated inner Tutte group ~(° u) is also finite. []

4. ORIENTATIONS OF PROJECTIVE SPACES OVER FIELDS

In this section we want to show how the inner Tutte group controls the

orientability of projective spaces over a field K of dimension at least 2. In case

K is formally real we shall characterize all orientations and in case K is not

formally real we shall recover the well-known fact that the corresponding

projective spaces are not orientable. Oriented matroids have been introduced in I-3]. An equivalent definition

has been given in [10] as well as in [4], where oriented matroids are viewed as

n-ordered sets or chirotopes.

D E F I N I T I O N 4.1. A chirotope (E, Z), of rank n is a set E together with a

map Z: E" --* {1, 0, - 1} satisfying:

(CO) There exist el, . . . , e. ~ E with z(el, . . . , e,) # O.

(C1) Z is alternating, i.e. for all el, . . . , e .~ E and all permutations n of

{1, . . . , n} we have

z(e~(1) . . . . . e~¢.)) = sgn n ' z ( e l . . . . . en).

(C2) I f w e { l , - 1}, eo, el . . . . ,en, f2 . . . . . f . e E a n d

w' ( - -1) i 'Z(eo . . . . , ei . . . . , e , ) .z(ei , f2 , . . . , f . ) >i 0

for all i = 0 . . . . . n,1

I As usual, the symbol (eo .. . . . el . . . . . e.) denotes the sequence (e o .. . . . ei- 1, ei+ 1 . . . . . e.) one gets from (e o .. . . . e.) by deleting ev

ON C O M B I N A T O R I A L AND P R O J E C T I V E G E O M E T R Y 189

then

Z(eo . . . . . g~i . . . . . e,)')~(ei, f2, . . , , f , ) = 0

for all i = 0 . . . . . n.

A chi ro tope (E, Z), induces a ma t ro id M = M z on E of rank n, whose bases

are all subsets {el . . . . . e,} __ E with z(el, . . . , e,) ¢ 0.

D E F I N I T I O N 4.2. An oriented matroid is an equivalence class of chirotopes (E, X)., where )fi, )~2: E" --. {1, 0, - 1} are called equivalent if either Z1 ~ ~(2 or

Zl ~ --X2"

D E F I N I T I O N 4.3. Two chirotopes (E, Z1), and (E, Z2), of rank n are said to

define projectively equivalent oriented matroids on E, if there exists some e { 1, - 1 } and some m a p t/: E ~ { 1, - 1 } such tha t for all ei . . . . . e, e E we

have

(4.1) Zl(ei, . . . , e,) = a- f i tl(ei)'z2(el . . . . , e,). i=1

The relat ion between chirotope structures and the Tut te group is summed up in

T H E O R E M 4.4. I f (E, Z), is a chirotope of rank n and M = M z, then Z induces a homomorphism, also denoted by X:

Z: T ~ -* {1, - 1}: Z(~M) := -- 1, Z(T(el . . . . . . . )):= •(el . . . . . e,)

such that for any four pairwise distinct hyperplanes H~, H2, H3, H 4 which intersect in a hyperline one has

H2

Vice versa, i f M is a matroid o f rank n, defined on E, and Z: - ~ ~ {1, - 1} is a

homomorphism with Z(eM) = -- 1, satisfying (4.2), then the associated map f rom

E" into {1, O, - 1 } , also denoted by Z and defined by

)~(e,, . . . , e ,):= ~x(T( . . . . . . . . . )) i f {el, . . . , e . } e ~ = ~ M (o otherwise

defines a chirotope on E of rank n.

Hence, given a matroid structure M o f rank n on E, there is a one-to-one correspondence between chirotopes (E,)0, with M z = M and homomorphisms

Z: ]]-mM ~ {1, -- 1} with Z(e~) = -- 1 and satisfying (4.2). Moreover, two such

homomorphisms define the same oriented matroid on E if and only if they


coincide on T M while they define projectively equivalent oriented matroids on E if and only if they coincide on T~ ).

Proof. This is Theorem 6.1 in [6], where the former condition (6.1) has been reformulated in terms of cross ratios, using [6, Th. 1.2]. (For a more thorough study of projective equivalence see also [16].) []

REMARK. [6, Th. 6.1] has been formulated only for finite E; however, the proof works also for infinite E.

For the rest of this section assume that M = M(~) is the projective space of dimension n - 1 >/2 over some field K, defined on E. Furthermore, assume that ( f n )n~r is a system of hyperplane functions for M and K and let qxT~ )¢-~K* denote the corresponding isomorphism given by

Proposition 1.5 and Theorem 3.6.

LEMMA 4.5. For any four pairwise distinct hyperplanes Hi, H2, H 3, H4

which intersect in a hyperline we have

H2 +~0 = 1 . (4.3) q~ H 3 H,, H 3 H 1

Proof. This is [6, Th. 3.1, (i) =¢, (iii)]. (The proof given there also works for infinite E.) []

Now, using the notations from Theorem 4.4, we can show

THEOR EM 4.6. Assume there exists some chirotope structure Z: En ~ {1, 0, - l} on M. Then K is formally real, and

(4.4) K + := {x~K* [ Z((p-i(x)) = 1}

is a domain of positive numbers in K. Proof. Let W : K * ~ { i , - 1 } denote the composition W:= Xotp - l . We

have qJ ( - 1) = Z(~M) = - 1 and thus K = K + w K - u {0} with

(4.5) K - : = { x E K * I V ( x ) = --1} = { x e K * I - -x~K+} .

Furthermore, K + is multiplicatively closed. It remains to show that for xl, x 2 ~ K + we have xl + x 2 e K +.

By (4.5) we have xi + x2 ~ 0. Put

y l :=Xl" (X i +x2) -1, y2:=X2"(Xi q-X2) - I .

Then we have Yl, y2eK*\{0, 1) and Yl + Y 2 - 1. By Theorem 3.5 and


Theorem 3.6 there exist four pairwise distinct hyperplanes H1, H2, H 3, H4 in M which intersect in a hyperline such that

(I ]) H2 . (4.6a) Yl = q~ Ha H4

Then (4.3) implies

(4.6b) Y2-- ~o Ha HI "

Now (4.2), (4.6a) and (4.6b) yield

+ z {0, 2). ~I/(yl) "1- tI/(y2) ----'Z H3 H4 Ha H i

Thus, there exists some i~ {1, 2} with ~(Yi) = 1. Since also q~(xi) = 1, we get

W(x1 + x2) = W(xl)-" Ug(yi)- 1 = 1

and thus xl + x 2 E K + (and, hence, V?(yj) = Ug(xj) = 1 for j~{1,2}\{i} as well!). []

Since, for obvious reasons, every matroid which is representable over some ordered field is also orientable, Theorem 4.6 implies (see also the proof of Theorem 4.8 below)

THEOREM 4.7. Assume M is the projective space of dimension n - 1 >1 2

over some field K. Then M is orientable i f and only if K is formally real. []

Finally, we show

THEOREM 4.8. Assume K is an ordered field and let M = M ( ~ ) denote the

projective space o f dimension n - 1 >>. 2 over K. Furthermore, assume

g: E ~ Kn\{0} is a representation of M and choose some system o f hyperplane

functions ( fH)H~e for M and K related to g; that is, for every i t ~ = ~f~u

there exists some base {all,1 . . . . . an,n-1} o f M IH and some Cn ~ K* such that

for all e ~ E we have

f , ( e ) = cn " det(a(an,1), . . . , g(au,~- O, a(e)).

-¢(o) c_~ K* denote the isomorphism induced by ( f n )H~r and assume that Let qg: ° M

z:E~-~ {1, 0, --1} defines a chirotope structure Z: T~M~{ 1, --1} on M

satisfying (4.4), that is ZIT~) = sign o tp. Then there exists some ~ ~ {1, -1} and

some map tl: E ~ {1, -1} such that for all el . . . . . e~EE we have

X(el . . . . . e~) = a. l~I t/(ei), sign det(g(el), . . . , g(en)). i=1


Proof. Define Zo: E" ~ {1, 0, - 1} by

Zo(el . . . . . . en) := sign det(g(el) . . . . , g(e,)).

Zo defines a chirotope structure Zo: T ~ { 1 , - 1 } , on M, because for eo,

e l , . . . , e,, f 2 , . . . , f , ~ E we have the Grassmann-PliJcker relations

( - 1) g" det(g(eo) . . . . . g(~,) . . . . . g(e,))" det(g(e~), g(f2), • . . , g(fn)) = 0 i=0

which means that Zo satisfies not only (CO) and (C1) but also (C2). In view of Definition 4.3 the assertion of Theorem 4.8 just claims that X and Xo are projectively equivalent; that is (cf. Th. 4.4) that, as homomorphisms from ~-~ into {1, - 1}, the two maps Z and Zo coincide on "~MDt°). Since XIT~ = sign o tp by assumption it is therefore enough to show that also

(4.7) Xol~-~, = sign o ~o

holds. So assume L = tr({e 1 . . . . . e,-2}) is a hyperline in M and al, a2, aa, a4 ~ E \ L are such that the four hyperplanes Hi := a(L • {ai}), 1 ~< i ~< 4, are pairwise distinct. Then [6, Prop. 3.1] yields

([ H1 H21) = det(g(el) .... , g (en -2 ) , g(al) , g(a3)) ~o H3 H4

x det(g(el) . . . . . a(e~-2), g(al), a(a,)) -~

x det(g(el) . . . . . #(e,_ 2), g(a2), g(a4))

x det(g(el) . . . . . #(e,-2), a(a~), g(a3)) - ~

and thus we have indeed

H2 sign (tP (IH13 H , I ) )

= Zo(el, . . . , e.-2, al, a3)" Zo(el, - . . , e , -z , al, a4)

x Zo(el . . . . . e . - 2 , a2, a4)'Zo(el, . . . , e . - a , a2, a3)

which proves (4.7) in view of Theorem 2.7. Since by assumption Z satisfies (4.4), it follows also that the homomorph-

isms ~(o, X: ~ ~ {1, - 1 } coincide on T ~ ). Thus by Theorem 4.4 the chirotopes (E, Xo)n and (E, Z), define projectively equivalent oriented matroids, so, our assertion follows from Definition 4.3. []

ON C O M B I N A T O R I A L AND P R O J E C T I V E G E O M E T R Y 193

5. A R E L A T I O N W I T H D I E U D O N N I ~ D E T E R M I N A N T S

Finally we want to connect our results with Dieudonn6 's construct ion of (non-commutat ive) determinants over skewfields (cf. [1, Ch. IV, §1]). T o this end we assume as before that M is a ma t ro id defined on E of finite rank n with

= ~M denot ing its set of bases, J f = ~¢~M its set of hyperplanes, and

a = a M its closure operator . Fur thermore , we denote by

(5.1) GM := {a: E ~ E [ a is bijective and for B ___ E we have

B e ~ if and only if ~(B) e ~ }

the automorphism group of the mat ro id M.

D E F I N I T I O N 5.1. Fo r a e GM put

(5.2) U(a) := (T~(a) • Ta -1" Tb" Z ~ I a, b e ~(M)) ~-~ "~-M (ZZ -~-~M

where, as before,

and, of course,

a ( a l . . . . . a . ) : = ( ~ ( a l ) , . . . , o~(an) ).

Then the determinant of a is defined as the coset

(5.3) det c~:= T~(a). T~ -1. U(~) for a e ~ M )

in T M.

R E M A R K S . (i) By Propos i t ion 2.14 and the exchange ax iom for bases we have, for ~ e GM,

(5.2a) U(e)

= (H(e).H(f)- 1.H'(~(f)).H'(e(e))- I lH ~ ,If, H':= e(H), e, f ~ E \H) .

(ii) Fo r ~, fl ~ GM, a e ~M) and b := fl(a) one has

T(~oflXa)" T ; 1 = (T~(b)" rb-1)" (Tfl(,) • r~- ' ) .

This implies

U(a o fl) c U(a)" U(fl) and det(~ o fl) ___ det a . det ft.

For U ~ < T M w e p u t

(5.4) G U := {c~ ~ GM [ U(cO <~ U }.

Remark (ii) yields at once


COROLLARY 5.2. For all U <~ T u the map det induces a well-defined homomorphism detv: G~ ~ -~ M/U given by

(5.5) detv(~t):= T~(a )" T ; 1" U for a ~ ( M ). []

REMARK. If U 1 ~< U 2 ~< T M, then we have G~ ' ~< G~t :, and the following diagram commutes where co: G~' c~ G~t: is the canonical imbedding and ~: Tu /U 1 --~ TM/U 2 denotes the canonical epimorphism.

G ~ ~ e, ot~,, " T u / U 1

G U2 detv~ :L -~M/U 2

For the rest of this section we assume that K is a skewfield and M is the left vector space K" for some n I> 3; that is E := K" and the matroid structure is given by linear (in)dependence. Moreover, assume that ((I)n)n~r is a fixed system of left linear forms for M with ~ 1({0}) = H for all H ~ g = ~ufu. We show

THEOR EM 5.3. The homomorphism q~:-~ ~ Kc, given by Proposition 1.5

and defined by

~o(eu) := - 1,

q)(Hu(a)):= ~u(a) for H ~ f , a t E \ H ,

induces an isomorphism ~0 o = q~[~-~): q[(~) c~ Kc. Proof. Let M' denote the projective space of dimension n - 1 over K

defined on E'. We may assume E' ~ E\{0} and #{2~K*I2"a~E '} = 1 for all a~K"\{O}. By Proposition 1.3 we have a well-defined homomorphism

fl: .,~' ,rg Tu, ~ T u given by

/3(~M'):= eM,

/3(Hu,(a)) := (r(H)u(a) for H ~ ~ ' : = ;It"u,, a ~ E'kH,

and /3 induces a homomorphism 13o: -u'-"-(°) ~ -u-"-(°). Then , . ~ ( 0 ) q~o.=q~oO/3o:.u,C-~Kc is an isomorphism by Theorem 3.6, and the

following diagram commutes:

T(o) ~ ~> Kc M'


In particular, fl0 is injective, and ¢Po is surjective, flo is also surjective by

Theorem 2.7, because we have

~u = {aM(H) I H • ~M'} .

Therefore flo and ¢Po must be isomorphisms. []

REMARK. The homomorphism tplL, is canonical in the sense that we would have obtained the same map if we had considered another family

( O , ' c u ) u ~ r of hyperplane functions for certain cn • K*.

In the sequel we put

(5.6) Uo := - r U n ker ~o.

Thus by Theorem 5.3 the following sequence of abelian groups is exact:

1 ~ Uo ~ TM"~Kc "~0,

-rto) In and T u / U o is canonically isomorphic to K c and thus also to - u -

particular, we have T M = T(M °°) x Uo; that is, YM = T(u °J" Uo and D-to) uc~Vo={1}.

Let G <~ Gu denote the group of all non-singular left linear transformations ~t: K" c~ K". We want to show that G ~ G v°, and we shall see that

detvo(~) corresponds to the (non-commutative) Dieudonn6 determinant for

all ct • G taking its values in K c.

Since from now on we shall only consider the matroid M we always write

H(a) instead of H u ( a ) for H • J ~ = ~ M and a • K " \ H .

P R O P O S I T I O N 5.4. G v° contains the group G of all non-singular left linear

transformations o~: K" ¢-~ K". Proof. Assume ~ • G. By (5.2a) and (5.4) we have to show that for H • ~ ,

H ' := ct(H), e, f • E \ H and

T : = n(e)" n ( f ) - 1. n '(~(f))" n'(~(e)) -1

we have ¢(T) = 1. Choose 2 • K* with e - 2 "f• H. Then we have also ~(e) - 2. ~(f) • H', and

thus we get

q~( T) = OH(e ) • OH( f ) - l " OH,(a(f) ) . OU,(a(e))- I

= 2 . 2 -1

- -5 . [ ]

Now we can show

196 A N D R E A S W. M. DRESS A N D W A L T E R W E N Z E L

T H E O R E M 5.5. Let (p: TM/U o c_~ Kc denote the isomorphism induced by ~p, and let deto: G--* Kc denote the Dieudonn~ determinant. Then the following diagram commutes:

G deta ) Kc

-gM/Uo

In particular, detvo is surjective. Proof. For a e G we have to show ~(detvo(a)) = detD(e). Choose some fixed

base B = {e 1 . . . . . en} of M. By [1, Th. 4.1] it suffices to consider the

au tomorphisms -(~) for 1 ~< i,j <~ n, i C j, 2 s K * and e~ for 2 ~ K * , defined by ~i,j

(5.7a) ~ l ~ ( e k ) : = e k for 1 ~< k ~< n, k :~j,

(5.7b) al,~)(ej) := e; + 2"ei;

(5.8a) az(ei):= ei for 1 ~< i ~< n - 1,

(5.8b) ~tz(e,) := ,~ • e,.

. (4) with i,j, ,~ as above we have detD(a) = 1. On the other hand, wc For a = ~i.j

get with e ) := ej + ;~'el and H : = a(B\{ej}) because of ei~H:

(p(detvo(a)) = q~(T(e ...... ej_~;,~j ....... e.)" T(~...,~))

--,~.(e)).,~.(ej)-'

N o w assume ~ = a~ for some ,~ ~ K*. Then we have detD(a) = ~, and we get

with e'. := 2 -e . and H := a(B\{e.}):

(p(detvo(a)) = q)(T(ew. . , e . . . . e'~)" T(et~...,e.))

= (I)H(e'.) " O H ( e . ) - 1

=i

which proves our Theorem. [ ]

When K is commutative, Theorem 5.5 also follows easily from [6, Prop. 3.1].

For a further study of determinants of au tomorphisms of matroids within the much more general and abstract f ramework of matroids with coefficients, see

[7, §4].


R E F E R E N C E S

1. Artin, E., Geometric Algebra, Princeton Univ. Press, Princeton, New Jersey, 1957. 2. Baer, R., Linear Algebra and Projective Geometry, Academic Press, New York, 1952. 3. Bland, R. G. and Las Vergnas, M., 'Orientability of matroids', J. Combin. Th. Ser. B 24

(1978), 94-123. 4. Dress, A. W. M., 'Chirotopes and oriented matroids', Bayreuth. Math. Schr. 21,

Tagungsbericht 2. Sommerschule Diskrete Strukturen, Bayreuth, 1985, pp. 14-68. 5. Dress, A. W. M., 'Duality theory for finite and infinite matroids with coefficients', Adv. in

Math. 59 (1986), 97-123. 6. Dress, A. W. M. and Wenzel, W., 'Geometric algebra for combinatorial geometries', Adv. in

Math. 77 (1989), 1-36. 7. Dress, A. W. M. and Wenzel, W., 'Grassmann-P1/icker relations and matroids with

coefficients' (to appear in Advances in Mathematics). 8. Dress, A. W. M. and Wenzel, W., 'Endliche Matroide mit Koeffizienten', Bayreuth. Math.

Schr. 26 (1988), 37-98. 9. Grundh6fer, T., 'The groups of projectivities of finite projective and affine planes', Ars

Combin. A25 (1988), 269-275. 10. Gutierrez Novoa, L., 'On n-ordered sets and order completeness', Pacific J. Math. 15 (1965),

1337-1345. 11. Hughes, D. R. and Piper, F. C., Projective Planes, Springer-Verlag, New York, Heidelberg,

Berlin, 1972. 12. Pickert, G., Projektive Ebenen, Springer-Verlag, Berlin, G6ttingen, Heidelberg, 1955. 13. Tutte, W. T., 'Lectures on matroids', J. Res. Nat. Bur. Standards B69 (1965), 1-47. 14. Welsh, D. J. A., Matroid Theory, Academic Press, London, New York, San Francisco, 1976. 15. Wenzel, W., 'A group-theoretic interpretation of Tutte's homotopy theory', Adv. in Math. 7~7_

(1989), 37-75. 16. Wenzel, W., 'Projective equivalence of matroids with coefficients' (to appear in J. Combin.

Theory A).

Authors' address:

A. W. M. Dress and W. Wenzel, Fakult~it ffir Mathematik, Universit/it Bielefeld, Postfach 8640, 4800 Bielefeld 1, F.R.G.

(Received, October 26, 1989)

on combinatorial and projective geometry

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