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Math, Z. i24, 315-318 (1972) 9 by Spri~ger-Verlag 1972

On Incidence Matrices of Finite Projective and Affine Spaces

WILLIAMM, KANTOR

It is welt-known that the rank of each incidence matrix of all points vs. all e-spaces of a finite d-dimensional projective or affine space is the number of points of the geometry, where 1 <_e<-d-1 (see [1], p. 20). In this note we shall generalize this fact: Theorem. Let 0 <-e < f <= - e - 1, and let Me, l be an incidence matrix of all d e-spaces vs. all f-spaces of PG(d, q) or AG(d, q). Then the rank of M~,f is the number of e-spaces of the geometry. We shall only prove the theorem in the case of AG(d, q). The projective case is similar and simpler. We note that the same proof shows that,./f 1 < e < f < d - e , an incidence mat~'i-r of all e-sets vs. all f-sets of a set of d points has rank

(de)

The relevant definitions are found in [t], w167 and 1,4. The dimension 1.3 of a subspace X of a projective space will be denoted dim(X). The empty subspace has dimension - 1,

Proof. Let E and F denote any e-space and f-space, respectively. Set

{~ (E, F) = if E c F if E r F.

For a suitable ordering of e-spaces and f-spaces we have M,, s = ((E, F)), Let R (E) denote the row of M~,r corresponding to the e-space E, Assume that there is a nontrivial dependence relation among the rows of M,, S. This may be assumed to have the form

R(E*)= ~ a(E)R(E)

E e: E*

(1)

for some e-space E*, where each a (E) is areaI number. Let F be the group of all collineations of AG(d, q) taking E* to itself. If e ~ F then (E~, F~)=(E, F).

316

W.M. Kantor:

Then (1) implies that, for all F,

(E*, e ) = (E*, ~ ) = 2 ~ (E~)(E ~, F ~)

E

= ~ a(E')(E, F),

E

so that R(E*)= ~

E,E*

a(E~)R(E). Thus,

Lrl R ( E * ) = ~ ~

a(E~)R(E)

F E=~E*

= ~ R(E) Z a(E~)9

E*E* F

(2)

Let ~ be the set of all ordered pairs (i,j) of integers satisfying the conditions - 1<i, j < e - 1 and j=i or i + 1. Order ~ lexicographically: (il,jO<(iz,ja) if either ia < i a or i1= i2 and Jl <J2. We now embed AG(d, q) in PG(d, q). Let H~ be the hyperplane at infinity. If (i, j)r ~, let o~(i,j) be the set of e-spaces E r H~ such that dim (E c~ E* c~ H~o)= i and d i m ( E n E*)=j. The g(i,j) are precisely the orbits of F of e-spaces other than E*. Fix E~fig(i,j). Then (2) implies that

IFIR(E*)=~ ~ R(E) ~ a(E~)

~(i,j) F

=~ ~ R(E)I~(i,j)[ ~a(E~j)

g(i,j) F

--Z [Ig(i,j)l Z a(E,5)] 2 R(E).

r g(i,j)

There is thus a dependence relation of the form

R(E*)= ~ bij ~ R(E)

with bij real. Let (m, n)e~. Since

r

(3)

e + f < d - 1 there is an f-space F~. such that

dim(Fm,nE*nHo~)=m and dim(Fm,~E*)=n.

As n < e - 1, Fro,z~E*. By (3), 0 = ~ b,j ~ (E, F,,,) whenever (m, n)~ ~. Let (i,j), (m,n)~. If (E, Fm,)=I for some E~g(i,j) then that i < m, j < n. In particular, if (i, j) > (m, n) then

g(i,j)

(4)

EcFmn implies

~] (E, F,,~

g(i,j)

(5)

On IncidenceMatrices of Finite Projectiveand AffineSpaces

317

Also, there is an e-space E c F~ such that E c~ E* c~ H~ = F~ c~ E* c~ H~ and E n E* = Fi~n E*. Thus,

(E, 5 ) * o.

g(i,j)

(6)

(7)

By (4) and (5), 0= ~

(i,j)<(m,n)

b~j ~ (E, fm~)

~(i,j)

for all (m, n) e ~. This is a system of l~l equations in the I~1 unknowns bij , (i, j)~ ~. If we use the ordering of ~, the coefficient matrix is triangular, with nonzero diagonal entries by (6). It follows that bi~=0 for all (i,j)e~, contradicting (3). This completes the proof. We note incidentally that the matrices Me, y have the following property: if

O<e<f<g<d-1,

then Me,fMy,g=G,y, g M~,g, where ce,y,g is the number off-spaces contained in a g-space G and containing an e-space contained in G. Corollary l. I f O < e < _ d - e - 1 , then a collineation of PG(d,q) induces similar permutations on the sets of e-spaces and ( d - e - 1)-spaces.

Proof. [1], p. 22.

Corollary 2. Let F be a collineation group of PG(d, q) or AG(d, q). I f

O<e<f<d-e-1, then F has at least as many orbits of f-spaces as e-spaces. Moreover, these quantities are equal in the projective case if f = d - e - 1. Proof. [1], pp. 20-22.

Corollary 3. Let F be a colIineation on f-spaces. I f O<e< f < d - e - 1 then on f-spaces is at least as large as that equal in the projective case if f = d - e -

group of PG(d, q) or AG(d, q) transitive F is transitive on e-spaces, and its rank on e-spaces. Moreover, these ranks are 1.

Proof. See the proof of [2], Theorem 4.4.

We remark that, by using [1], p. 21, together with the analogue of our theorem for incidence matrices of subsets of a finite set, we also obtain an elementary proof of Theorem 1 of Livingstone and Wagner [3]. Parts of these corollaries are due to Wagner [5] in the projective case.

Added in Proofs. The corollaries can also be easily deduced from the following facts. Let 0e be the permutation character obtained from the action of the full collineation group of PG(d, q) or AG(d, q) on the e-spaces of the geometry, where O < - e < d - 1 . If O<-e<d/2, then 0~_lc0e=0d_l_ ~ for PG(d,q), and 0~_l~0d_e~0 ~ for AG(d,q). These inclusions can be proved by calculating (0e, Of) whenever O < e < f < d - 1 . In the projective case, this result and this calculation are essentially contained in [4], pp. 276-278.

318

W.M. Kantor: On Incidence Matrices of Finite Projective and Affine Spaces

References

1. Dembowski, P.: Finite geometries. Berlin-Heidelberg-New York: Springer 1968. 2. Kantor, W. M.: Automorphism groups of designs. Math. Z. 109, 246-252 (1969). 3. Livingstone, D., Wagner, A.: Transitivity of finite permutation groups on unordered sets. Math. Z. 90, 393-403 (1965), 4. Steinberg: R.: A geometric approach to the representations of the full linear group over a Galois field. Trans. Amer. Math. Soc. 71, 274-282 (1951). 5. Wagner, A.: Collineations of finite projective spaces as permutations on the sets of dual subspaces. Math. Z. 111, 249-254 (1969). Prof. William M. Kantor Department of Mathematics University of Oregon Eugene, Oregon 97403 USA

(Received September 30, 1970)

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