- Email: [email protected]
Partial λ-Geometries of Small Nexus
Annals of Discrete Mathematics 6 (1980) 19-29 @ North-Holland Publishing Company
PARTIAL A-GEOMETRIES OF SMALL NEXUS Peter J. CAMERON Merton College, Oxford OX1 450, England
David A. D R A K E Universify of Florida, Gainesville, F L 3261 I, U.S.A.
In a partial A-geometry, each two points are joined by 0 or A blocks; each two blocks have 0 or A points in common; the block size k is constant; and for each nonflag ( p , G), there are precisely e blocks X with p in X such that X n G is not empty. Generalized quadrangles are partial 1-geometries with nexus e = 1. If A = 2, then e z=3; and the first author has determined that partial 2-geometries with nexus 3 exist precisely for the values k = 3,4, 8,24. We prove (1) that if A > 2 and k > e + 1, then e > 2A; (2) that A = 3, e = 7 implies k is one of 7, 15, 21, 24 or . A. There are 36. We call a partial A -geometry extremal if IG nH nK (> 1 implies (GnH nK ( > no extremal partial A-geometries with e C A’- A. Such a geometry with e = A’- A + 1 and k > e is called a A-quadrangle. We determine all A-quadrangles with A > 2. They are constructed from quadratic forms of Witt index 4 on finite 8-dimensional vector spaces.
0. Introduction
In Section 1, we observe that the block graph of a partial A-geometry is strongly regular and apply the rationality-integrality conditions for strongly regular graphs to obtain a non-existence criterion for partial A-geometries. In Section 2, we obtain a lower bound on the nexus of a partial A-geometry. In Section 3 , we obtain a better lower bound for the nexus e of the subclass of extremal partial A-geometries; namely, e > A’- A. A proper extremal partial A-geometry with e=A2-A+1 is called a A-quadrangle. In the remainder of Section 3, we determine the A-quadrangles with A > 2. (Those with A = 2 have already been determined up to possible non-uniqueness when k = 24.) To obviate excessive repetition of the word “finite”, we now assert that all incidence structures considered in this paper are tacitly assumed to be finite. A la Dembowski, we write [ p , , . . . , p,,] to denote the number of blocks which contain the point set { p l , . . . ,p,,}, [G,, . . . , G,,] for the dual notion. 1. The block graph of a partial A-geometry
Definition 1.1. For A > 0, a partial A-geometry (with nexus e > 0) is an incidence structure with b blocks and v points which satisfies: (i) [ p , q ] = 0 or A for each point pair ( p , q ) with p f q ; 19
P.J. Cameron and D.A. Drake
20
(ii) [G, H]= 0 or A for each block pair (G, H ) with G # H ; (iii) for each non-incident point-block pair ( p , G), there exist precisely e blocks X with p E X and [ X , GI # 0; (iv) A < [ P I < b for every p, and A <[GI< u for every G ; if A = 1, we also assume the existence of integers k, r such that [GI = k and [ p ] = r for all G, p . Partial 1-geometries were first studied by Bose who called them simply “partial geometries”. He proved the following result in the special case that A = 1 (see [2, p. 3981). The result with A # 1 is due to the second author [6, Lemma 1.31.
Lemma 1.2. Let 3 be a partial A-geometry. Then there are integers k , r such that [GI = k and [ p ] = r for all G, p. Further,
v = [ k ( r- l ) ( k - A)/eA]+ k,. b=[r(r-l)(k-A)/eA]+r.
If A f 1 , r = k ; and therefore b = v. If e = k, each pair of points in a partial A-geometry 54 is joined, hence % is a (balanced incomplete) block design. If e < k , we shall call % a proper partial A-geometry. A valuable tool in our study is the notion of the block graph 93 of an incidence structure 5 4 : 93 is obtained by using the blocks of 54 as vertices and taking two blocks to be adjacent if and only if they have a nonempty intersection. Recall that a strongly regular graph is a graph with the following properties: (1)each vertex is adjacent to the same number a of vertices; (2) 0 < a < n - 1 where n denotes the total number of vertices; ( 3 ) the number of vertices adjacent to both of G and H is a constant c if G and H are themselves adjacent, otherwise is the constant d.
Proposition 1.3. Let % be a proper partial A-geometry with A # 1. Then the block graph of 3 is strongly regular and has the following parameters: n = u = [ k ( k - l ) ( k - A)/eA]+ k, a = k ( k - 1)/A, c = [kA - k
d
-
A + e ( k -A)]/A,
= ek/A.
Proof. The value for n is given by Lemma 1.2. To compute n, however, one need only fix a block G and count (in two ways) the double flags (x,y, 2 )with x in G, y not in G. To compute d, one fixes a pair of non-intersecting blocks B and G and observes that the number of double flags (x, y , 2 ) with x in B and y in G is keA = dA2. Next, let B and G be fixed blocks with a non-trivial intersection; let f denote the number of blocks which have non-trivial intersection with B but are
Partial A-geometries of small nexus
21
parallel to G. Counting the flags (x, Y ) such that X E B -G, [Y,G ] = 0 , one obtains ( k - A)( k - e ) = fA. Clearly, c = a - 1- f. Now one easily computes a, hence c. Using the computed values for a and n and assuming that a = n - 1, one sees that k must be either e or A, a contradiction. The following theorem yields the so-called “rationality conditions” or “integrality conditions” for strongly regular graphs. For a proof, one may consult Cameron and Van Lint [5, pp. 15, 161.
Theorem 1.4. The adjacency matrix of a strongly regular graph has three eigenvalues whose multiplicities are 1 and
1
i * [ n - l * ( n - l ) ( d - c ) - 2a 2 ((d-~)‘+4(a-d))~’’
Proposition 1.5. Let 53 be a proper partial A-geometry with A # 1. Then the following expression is an integer: k2(k-l)(k-A) eA[k ( A + 1)- A(e + l)]. Proof. Apply Proposition 1.3 and Theorem 1.4, using the plus-sign in the expression in Theorem 1.4.
Proposition 1.6. Let 53 be a proper partial A-geometry with A # 1. Then p ( k ) = k ( A + 1 ) - h ( e + 1) is a nonzero integer which evenly divides the integer A3(e+ l)’(he - l ) ( e- A ) . Proof. By Proposition 1.5, p(k) divides k’(k - l)(k - A). The desired conclusion follows from the following facts: GCD(p(k), k ) divides A(e + 1); GCD(p(k), k - 1) divides ( k - l)(A + 1)- p ( k ) = he - 1; and GCD(p(k), k - A ) divides ( k - A ) (A+l)-p(k)=A(e-A). Corollary 1.7. For each pair ( A , e ) with A > 1, e > 0, there are only finitely many proper partial A-geometries of nexus e.
Proof. By Proposition 1.6, there are only finitely many possible k’s for a given pair (A, e ) ; hence, by Proposition 1.3, only finitely many possible v’s. 2. A lower bound for the nexus of a partial A-geometry Generalized quadrangles are partial 1-geometries of nexus 1. Thus there are many partial A-geometries which satisfy A = e = 1; however, A # 1 clearly implies
22
P.J. Cameron and D.A. Drake
that e > A . The first author has investigated the situation when A = 2 and e = 3, obtaining the following
Theorem 2.1. (Cameron [4, Theorem 5.121). A partial 2-geometry with nexus 3 exists for precisely the values k = 3, 4, 8 and 24. With the possible exception of k = 24, the structure is determined by k. We shall devote Section 2 of this paper to proving
Theorem 2.2. Assume the existence of a partial A-geometry % with A > 2 and k > e + l . Then e>2A; and if A is even, e > 2 A + 1 . Remark 2.3. The requirement k > e + 1 is necessary: the existence of a Hadamard matrix of order 2A > 3 implies the existence of a partial A-geometry with k = 2 A and e = 2A - 1 (see [6, Proposition 2.3 and Lemma 1.81). Proof of Theorem 2.2. To begin, let {G, H } be a fixed pair of intersecting blocks of 3. Let ci be the number of blocks Y such that Gf Y f H , IG n YI >0, IH n Y1> 0 and 1 G n H n YI = i, 0 s i s A. Then the formula for c in Proposition 1.3 yields the equation
2 Ac,= k ( e + A - 1 ) - A ( e + l ) . A
t=O
Counting flags (x, Y) such that
XE
G n H and Hf Y f G, one obtains
A
1 ici = A(k -2).
(2)
i=O
Counting double flags (x, y, 2 ) such that x , y ~ G n H x, f y and G f Z f H , one gets
2 A
i=O
i(i-1)ci=A(A-1)(A-2).
(3)
We now multiply (1) by (j’+ j) where j is some integer, (2) by -2jA, and (3) by A, and add the three resultant equations. The left-hand side of this linear combination is 1 f(i)c, where
f(i) = i2A - i(2jA + A )
+ (j’A + jh).
Then f’(i) = 2iA -(2jA + A ) = 0 when i = j + f , and f”(i) = 2A > O ; so f(i) has an absolute minimum at i = j ++. Since f(j) = f ( j+ 1)= 0, f(i) 3 0 for all integral values of i. Since the ci’s are also all non-negative, the left-hand side of the linear combination of (1)-(3) is non-negative; and O S ( j ’ + j)[k(A + e - 1)- A(e + 1)]- 2jA(kA -2A)
+(A4-
3A3 .t 2A2).
Partial A-geometries of small nexus
23
then kj[2A2- ( j + l ) ( h+ e - 1 ) ] GA 4 - 3h3+ 2A2+4jh2- h ( e + l)(j”+j).
(4)
Then if 2A2- (j + l ) ( h+ e - 1 ) > 0 , we may replace k by ( e + 1) in (4), obtaining a strict inequality. Restating, we have
( e + 1)[2jh2- (j’
+ j ) ( A + e - l ) ]
-
hej” - hej - Aj” - A j
(5)
provided that 2A2-h-e+l >j . h+e-1 Now (5) simplifies to
O<(h4- 3h3+ 2h2)+j(-2eh2+ 2h2+e 2 - 1 ) + j 2 ( e 2 -1).
(7)
Consider the quantity d
+ 1).
=h2/(e
(8)
We wish to substitute [ d ] , the largest integer
(2A2- h - e + l ) ( e+ 1 ) > h2(h+ e - 1).
(9)
This inequality is equivalent to O>e2+e(-A2+A)+(A3-3h2+X - 1 ) - h ( e ) .
(10)
Now h(e)=O if and only if 2 e = h 2 - A + ( h 4 - 6 h 3 + 1 3 h 2 - 4 A + 4 ) 1 ’ 2 . Since (h2-3h+2)2 3 , then inequality (10) holds when A - 1 e 2h + 1. Now denote the right-hand side of inequality (7) by g ( j ) , regarding e and h > 3 as fixed. Since j = d satisfies (6), O < g ( j ) for every integer j c d . Now g ’ ( j ) = j(2e2-2)-2eh2+2h2+e2- 1,
so g’(j) = 0 if and only if
2d-1 j = 2eh2-2h2-e2+1 -2e2- 2
2
.
Since g”(j) = 2e2- 2 > 0 , g ( j ) is a declining function to the left of j = i(2d - 1) and a rising function on the right. We wish to prove that g ( d - 1 ) = g ( d ) < O . It will follow that g ( [ d ] ) < O , violating (7) and proving the non-existence of a partial A-geometry with nexus e. One easily computes g ( d - 1 ) = g(d)+2eh2-2h2- e2+1 + ( 1 -2d)(e2- 1 ) = g ( d ) .
P.J. Cameron and D.A. Drake
24
Next, one computes
( e + l)g(d)/A2=2A2-3A(e+l ) + ( e + 1 ) 2 and concludes that g(d) <0 if A - 2 < e < 2A. We have proved
Proposition 2.4. There are no proper partial A-geometries whose parameters satisfy A > 3 . k > e + l and e<2A.
Then (6) is satisfied by j = 1 for all e <7. Setting A = 3 and < e < 7. Next, assume the existence of a partial A-geometry with parameter e = 2A. Then (7) simplifies to Now, assume A
= 3.
j = 1 in (7) yields a contradiction if A
0<(A4-3A3+2A2) +j(-4A3+6A2- 1)+j2(4A2-1).
(11)
The preceding inequality must hold when j satisfies (6); i.e., when
2A2-3A
+ 1> j(3A
1).
(12) Assume that A is an even integer. Then j=i(A-2) satisfies (12), hence may be substitued into (11) to yield O<-A2+2A, a contradiction. If A is odd and if e = 2A, one observes that j =;(A - 1) satisfies (12); then one may substitute j = ;(A - 1) into (11) to obtain the contradiction 0 < -A2 + 1. Lastly, assume that e = 2A + 1, A even. Then (7) becomes -
O<(A4-3A3+2A2)+ j(-4A3+4A2+4A)+ j2(4A2+4A). (13) The preceding inequality must hold when j satisfies (6); i.e., when 2A2- 3A > 3jA. Then (13) must hold when j =+(A - 2); this requirement yields another contradiction and thus completes the proof of Theorem 2.2. We conclude this section by considering the possible 3-geometries of nexus 7. Proposition 1.6 implies that 4k - 24 divides 2'' . 33 . 5 evenly. Then 4k - 24 = 2" . 3' . 5' where a < 10, b <3, c < 1. Since 4k - 24 is a multiple of 4, a 3 2. By Proposition 1.3, the integer d = ek/A =:k, so 3 I k. Then 4k -24 is a multiple of 3, so b 3 1. By Proposition 1.3, the integer u - k =&k(k - l ) ( k -3), so 7 I [k(k - l ) ( k -3)]. Then k '0, 1 or 3 (mod 7), and k = 2a-2. 3b . 5'+6, so
2"-2 * 3'
*
5' = 1, 2 or 4
(mod 7).
Thus 3b * 5" = 1, 2 or 4 (mod 7), and hence 3' . 5' is one of 9, 15, 135. We have proved that k = 6 + a . 2 @where a = 9 , 15 or 135 and O S p S 8 . Since A = 3 and e = 7, the integer in Proposition 1.5 is Dk =
k4-4k3 + 3 k 2 21(4k -24)
( k - 6)4+20(k - 6)3+ 147(k - 6)2+468(k - 6) + 540 84(k - 6) a 4 . 24p + 2oa3 . 23p + 1 4 7 .~ 220 ~ +46sa . 2 @+ 540 21a . 2 p + 2
Partial A-geometries of small nexus
25
+
If p 2 2, 2p 3 p 2; so 2p+2I 540, and hence p =s1. Then the only possible values of k are 15, 24, 21, 36, 141 and 276. One can see that Dk is not an integer for the two largest of these values, so we have proved
Proposition 2.5. If 3 is a partial 3-geometry with nexus 7, then the block size in 59 must be 7, 15, 21, 24 or 36. 3. A-quadrangles In this section, we define A-quadrangles. For A = 2, these are simply the proper partial A-geometries of nexus 3 discussed in Theorem 2.1. We will completely determine the A-quadrangles with A > 2.
Definition 3.1. A partial A-geometry will be called extremal if the condition lGnHHnKI>1 for blocks G, H, K implies that (GHnHflKIaA. Proposition 3.2. The inequality e 2 A* - A + 1 holds in every extremal partial A-geometry 3. Equality holds if and only i f the following criterion is satisfied: for each nonf7ag ( p , G), the e blocks through p which intersect G induce a projective plane of order A - 1 on the e points of G which are joined to p (degenerate unless A >2). Proof. Let ( p , G) be a nonflag, S be the set of e points of G which are joined to p . Since 3 is extremal, the blocks through p induce a “partial plane” E on S ; i.e., no two points of S lie in more than one common induced block. The number of pairs of points of S which are joined in E is ieA(A - 1) while the total number of pairs of points of S is + e ( e - I). Definition 3.3. A A-quadrangle is a proper extremal partial A-geometry with e = A * - A + I. We describe next a family of A-quadrangles. We refer to [l, 7, 81 for the geometric background: see in particular [l, Theorems 3.8 and 3.9 (Witt’s Theorem)].
Example 3.4. Let V be an %dimensional vector space over the field GF(q). Let Q be a quadratic form of Witt index 4 on V. The totally singular subspaces of dimension 4 will be called half spaces. The half spaces can be divided into two families with the property that, for any two half spaces S and T, the codimension of S n T in S is even if and only if S and T belong to the same family. (This is implicit in [8, 7.12 and 8.4.31; we give a short direct proof in the Appendix.) Define an incidence structure %(q)as follows: the points of %(q) are the totally
26
P.J. Cameron and D.A. Drake
singular 1-dimensional subspaces of V; the blocks are the members of one family of half spaces; and incidence is the relation of containment.
Proposition 3.5. % ( q ) is a ( q + 1)-quadrangle with e = q2 + q + 1, k q2+q+l.
= q3
+
Proof. Any block, with the points it contains, is a projective 3-space over GF(q); and two blocks meet in the empty set or a line. Given a totally singular 3-dimensional subspace U = ( u l , u2, u,), Witt’s Theorem yields a basis {ul,. . . , u4, w l , . . . , w4} for V consisting of singular vectors with B(u,,w , ) = 1, B(u,,u,) = B(u,, w,) = B(w,, w,) = 0 for i f j (where B is the bilinear form associated with 0).If follows that ( u l , u2, u,, u4) and ( u l , u2, ug, w4)are the only half spaces containing U and that exactly one of these is a block. Counting arguments (which we omit) now show that any point lies in 4 , + q2 + q + 1 blocks and that any two points on a singular line L lie in q + l blocks whose intersection is L. Of course, two points on a hyperbolic line cannot lie in a block. Finally, if S is a half space and p = ( u ) a point not in S, the map w -+ B(u, w ) is a nonzero linear form on S. A point incident with S is contained in the kernel of this map if and only if it is joined to p by a singular line. Then p is joined to q 2 + q + 1 points of S. It follows that the same number q 2 + q + 1 of blocks containing p meet S. This completes the proof. Our main result is the converse to Proposition 3.5
Theorem 3.6. A A-quadrangle with A 3 3 is isomorphic to %(q) where A
=q
+ 1.
Proof. We let % be a A -quadrangle with A 3 3. Put A = q + 1, hence e = q2+ q + 1. A non-empty intersection of two blocks will be called a line. If ( p , G) is a non-flag, the set n(p,G) of points of G collinear with p will be called a plane. Proposition 3.2 shows that a plane, equipped with the lines it contains, is a projective plane (of order 4 ) . It is clear that any two points of a block lie in a unique line, and we prove
Lemma 3.7. Three non-collinear points of a block lie in a unique plane. Proof. Let p , q, r be non-collinear points of G. There is a block H f G containing p and q ; r is joined to a point s of H outside G (indeed, to q2 such points); and p , q, r lie in the plane n(s,G). This plane is the smallest linear subspace containing p , q and r, hence is uniquely determined by these points. It follows from the classical axioms of projective geometry that a block, together with its points, lines and planes, is a projective space. We let d - 1 denote its dimension, so that k = (qd - l)/(q - 1).
Partial h-geometries of small nexus
21
Lemma 3.8. d is a multiple of 4.
Proof. Let G and H be disjoint blocks. Define a relation - on G by the rule p q if either p = q or there is a block containing p and q and meeting H. This relation is clearly reflexive and symmetric; we show it is transitive. Suppose p , q and r are distinct points with p q r. Let X and Y be blocks containing p , q and q, r, respectively, and meeting H. Then X n H and Y f l H are lines of the plane IT(q, H ) ; so there is a point s in X n Y n H. Then p, q, r lie in the plane n ( s , G); so there is a block containing p, r and s, whence p r. Thus is an equivalence relation. Clearly every equivalence class contains the line through any two of its points, hence is a subspace of the projective space G. The number of lines through a point p lying in the equivalence class is equal to the number of lines in the plane IT(p, H ) , that is, q 2 + q + 1; so the size of an equivalence class is q3+ q2+ q + 1. Since G is partitioned into equivalence classes, (q4- l)/(q - 1) divides (qd - l)/(q - l), whence 4 divides d .
-
- -
-
Lemma 3.9. d
-
= 4.
Proof. Proposition 1.6 asserts that k ( h + l)-A(e+ 1) divides- h 3 ( e+ l)’(Ae- 1) ( e -A). Substituting the values of A, e and k, we conclude that D(q, d) = (q + 2)(qd-4+qdp6+*
. .+ 1) - 1 divides
F ( q ) = ( q + 1)2(q2+q+2)2(q2+2q+2).
If q > 2 , each of (4’1)’ and (q2+q+2) is 2. Since F ( 2 )= 2’ . 45 <213, d 14 in general. By Lemma 3.8, d must be 4, 8 or 12. Supposing that d = 12, we denote D(q, 12) by D. Use of the Euclidean algorithm shows that (q + 1, D) = (q + 1,4), (q2+ q + 2, D ) = (q - 45,518) and (q2+ 2q + 2, D ) = (q + 64,397). Then D must divide 4’ . 5 1S2 . 397 = N. Certainly D = D(q, 12) is too large to divide N if q 3 300. Then we may assume q < 300; so (q + 64,397) = 1, and D must divide N/397 = 26 . 7’ 37’ = M. If q 3 7, then D > 7 9 > M, so q G 5 . One easily eliminates the remaining possibilities for q, so d ~ 8 One . eliminates the possibility that d = 8 by a shorter calculation of the same type.
-
Lemma 3.10. The geometry of points and lines is a polar space. Proof. We verify the axioms of Buekenhout and Shult [3]. It is clear that any line has at least three points and that no point is collinear with all others. Let L be a line and p a point not on L. We must show that p is collinear with one or all points of L. Let G be any block containing L. If P E G, then p is collinear with every point of L ; so suppose not. Then the points of L collinear with p are the
28
P.J. Cameron and D.A. Drake
members of cl(p, G) nL ; this set is a point or a line, since G is a projective 3-space. The rank of the polar space is four, since a block is a maximal totally singular subspace. Comparing the number of points of % (Lemma 1.2) with those in each of the six types of finite polar spaces of rank 4, we see that it is derived from a quadratic form of Witt index 4 on an 8-dimensional vector space; that is, it is of type D4(q)(see [8, p. 2191). Since the number of blocks is half the total number of half spaces and since two blocks meet in the empty set or a line, the blocks must comprise one family of half spaces. The theorem is proved.
Remark. Another way of making the identification is to observe that the only candidates for maximal totally singular subspaces containing a plane 17 = n(p,G) are the unique block containing 17 and union of the lines joining p to points of 17. Thus each plane lies in just two maximal subspaces. Now apply [8, 7.12 and 8.4.31.
Appendix Let V be a vector space of dimension 2 n over a commutative field, Q a quadratic form of Witt index n, €3 the associated bilinear form. As observed in the proof of Proposition 3.5, a totally singular subspace of dimension n - 1 lies in exactly two half spaces. Form a graph whose vertices are half spaces, two vertices adjacent if their intersection has dimension n - 1. Let S be a vertex, {TI,T2) an edge of the graph. Put T Ifl T2= U, and let d = dim(S n U ) .The functions s -+ B(s, u ) , u E U, form an ( n - d - 1)-dimensional subspace of the dual space S ’ ; its annihilator V is thus a (d + 1)-dimensional subspace containing S n U. Then (U, V) is a half space containing U, whence ( U , V) = TI or T2,say ( U , V) = T,. Then dim(S n T2)= dim(S f l T,)- 1. It follows that the distance function in the graph is given by the rule d ( S , T) = n -dim(S fl T), and that there are no circuits of odd length, that is, the graph is bipartite. Vertices in the same bipartite block lie at even distance. The claim made in Example 3.4 is proved.
Acknowledgment The second author wishes to thank Mark Hale for helpful discussions on symplectic geometries.
Partial A -geometries of small nexus
29
References [ l ] E. Artin, Geometric Algebra, Tracts in Pure and Applied Math. 3 (Interscience Publ., New York, 1957). [2] R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J . Math. 13 (1963) 389-419. [3] F. Buekenhout and E.E. Shult, On the foundations of polar geometry, Geometriae Dedicata 3 (1974) 155-170. [4] P.J. Cameron, Parallelisms of Complete Designs, London Math. SOC.,Lect. Note Series 23 (Cambridge Univ. Press, Cambridge, 1976). [S] P.J. Cameron and J.H. van Lint, Graph Theory, Coding Theory and Block Designs, London Math. SOC.,Lect. Note Series I9 (Cambridge Univ. Press, Cambridge, 1975). [6] D.A. Drake, Partial A -geometries and generalized Hadamard matrices over groups, to appear. [7] B. Huppert, Geometric algebra, Lecture notes, Dept. Math., Univ. of Illinois at Chicago Circle, Chicago, IL (1970). [XI J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 386 (Springer Verlag, Berlin, 1974).
PARTIAL A-GEOMETRIES OF SMALL NEXUS Peter J. CAMERON Merton College, Oxford OX1 450, England
David A. D R A K E Universify of Florida, Gainesville, F L 3261 I, U.S.A.
In a partial A-geometry, each two points are joined by 0 or A blocks; each two blocks have 0 or A points in common; the block size k is constant; and for each nonflag ( p , G), there are precisely e blocks X with p in X such that X n G is not empty. Generalized quadrangles are partial 1-geometries with nexus e = 1. If A = 2, then e z=3; and the first author has determined that partial 2-geometries with nexus 3 exist precisely for the values k = 3,4, 8,24. We prove (1) that if A > 2 and k > e + 1, then e > 2A; (2) that A = 3, e = 7 implies k is one of 7, 15, 21, 24 or . A. There are 36. We call a partial A -geometry extremal if IG nH nK (> 1 implies (GnH nK ( > no extremal partial A-geometries with e C A’- A. Such a geometry with e = A’- A + 1 and k > e is called a A-quadrangle. We determine all A-quadrangles with A > 2. They are constructed from quadratic forms of Witt index 4 on finite 8-dimensional vector spaces.
0. Introduction
In Section 1, we observe that the block graph of a partial A-geometry is strongly regular and apply the rationality-integrality conditions for strongly regular graphs to obtain a non-existence criterion for partial A-geometries. In Section 2, we obtain a lower bound on the nexus of a partial A-geometry. In Section 3 , we obtain a better lower bound for the nexus e of the subclass of extremal partial A-geometries; namely, e > A’- A. A proper extremal partial A-geometry with e=A2-A+1 is called a A-quadrangle. In the remainder of Section 3, we determine the A-quadrangles with A > 2. (Those with A = 2 have already been determined up to possible non-uniqueness when k = 24.) To obviate excessive repetition of the word “finite”, we now assert that all incidence structures considered in this paper are tacitly assumed to be finite. A la Dembowski, we write [ p , , . . . , p,,] to denote the number of blocks which contain the point set { p l , . . . ,p,,}, [G,, . . . , G,,] for the dual notion. 1. The block graph of a partial A-geometry
Definition 1.1. For A > 0, a partial A-geometry (with nexus e > 0) is an incidence structure with b blocks and v points which satisfies: (i) [ p , q ] = 0 or A for each point pair ( p , q ) with p f q ; 19
P.J. Cameron and D.A. Drake
20
(ii) [G, H]= 0 or A for each block pair (G, H ) with G # H ; (iii) for each non-incident point-block pair ( p , G), there exist precisely e blocks X with p E X and [ X , GI # 0; (iv) A < [ P I < b for every p, and A <[GI< u for every G ; if A = 1, we also assume the existence of integers k, r such that [GI = k and [ p ] = r for all G, p . Partial 1-geometries were first studied by Bose who called them simply “partial geometries”. He proved the following result in the special case that A = 1 (see [2, p. 3981). The result with A # 1 is due to the second author [6, Lemma 1.31.
Lemma 1.2. Let 3 be a partial A-geometry. Then there are integers k , r such that [GI = k and [ p ] = r for all G, p. Further,
v = [ k ( r- l ) ( k - A)/eA]+ k,. b=[r(r-l)(k-A)/eA]+r.
If A f 1 , r = k ; and therefore b = v. If e = k, each pair of points in a partial A-geometry 54 is joined, hence % is a (balanced incomplete) block design. If e < k , we shall call % a proper partial A-geometry. A valuable tool in our study is the notion of the block graph 93 of an incidence structure 5 4 : 93 is obtained by using the blocks of 54 as vertices and taking two blocks to be adjacent if and only if they have a nonempty intersection. Recall that a strongly regular graph is a graph with the following properties: (1)each vertex is adjacent to the same number a of vertices; (2) 0 < a < n - 1 where n denotes the total number of vertices; ( 3 ) the number of vertices adjacent to both of G and H is a constant c if G and H are themselves adjacent, otherwise is the constant d.
Proposition 1.3. Let % be a proper partial A-geometry with A # 1. Then the block graph of 3 is strongly regular and has the following parameters: n = u = [ k ( k - l ) ( k - A)/eA]+ k, a = k ( k - 1)/A, c = [kA - k
d
-
A + e ( k -A)]/A,
= ek/A.
Proof. The value for n is given by Lemma 1.2. To compute n, however, one need only fix a block G and count (in two ways) the double flags (x,y, 2 )with x in G, y not in G. To compute d, one fixes a pair of non-intersecting blocks B and G and observes that the number of double flags (x, y , 2 ) with x in B and y in G is keA = dA2. Next, let B and G be fixed blocks with a non-trivial intersection; let f denote the number of blocks which have non-trivial intersection with B but are
Partial A-geometries of small nexus
21
parallel to G. Counting the flags (x, Y ) such that X E B -G, [Y,G ] = 0 , one obtains ( k - A)( k - e ) = fA. Clearly, c = a - 1- f. Now one easily computes a, hence c. Using the computed values for a and n and assuming that a = n - 1, one sees that k must be either e or A, a contradiction. The following theorem yields the so-called “rationality conditions” or “integrality conditions” for strongly regular graphs. For a proof, one may consult Cameron and Van Lint [5, pp. 15, 161.
Theorem 1.4. The adjacency matrix of a strongly regular graph has three eigenvalues whose multiplicities are 1 and
1
i * [ n - l * ( n - l ) ( d - c ) - 2a 2 ((d-~)‘+4(a-d))~’’
Proposition 1.5. Let 53 be a proper partial A-geometry with A # 1. Then the following expression is an integer: k2(k-l)(k-A) eA[k ( A + 1)- A(e + l)]. Proof. Apply Proposition 1.3 and Theorem 1.4, using the plus-sign in the expression in Theorem 1.4.
Proposition 1.6. Let 53 be a proper partial A-geometry with A # 1. Then p ( k ) = k ( A + 1 ) - h ( e + 1) is a nonzero integer which evenly divides the integer A3(e+ l)’(he - l ) ( e- A ) . Proof. By Proposition 1.5, p(k) divides k’(k - l)(k - A). The desired conclusion follows from the following facts: GCD(p(k), k ) divides A(e + 1); GCD(p(k), k - 1) divides ( k - l)(A + 1)- p ( k ) = he - 1; and GCD(p(k), k - A ) divides ( k - A ) (A+l)-p(k)=A(e-A). Corollary 1.7. For each pair ( A , e ) with A > 1, e > 0, there are only finitely many proper partial A-geometries of nexus e.
Proof. By Proposition 1.6, there are only finitely many possible k’s for a given pair (A, e ) ; hence, by Proposition 1.3, only finitely many possible v’s. 2. A lower bound for the nexus of a partial A-geometry Generalized quadrangles are partial 1-geometries of nexus 1. Thus there are many partial A-geometries which satisfy A = e = 1; however, A # 1 clearly implies
22
P.J. Cameron and D.A. Drake
that e > A . The first author has investigated the situation when A = 2 and e = 3, obtaining the following
Theorem 2.1. (Cameron [4, Theorem 5.121). A partial 2-geometry with nexus 3 exists for precisely the values k = 3, 4, 8 and 24. With the possible exception of k = 24, the structure is determined by k. We shall devote Section 2 of this paper to proving
Theorem 2.2. Assume the existence of a partial A-geometry % with A > 2 and k > e + l . Then e>2A; and if A is even, e > 2 A + 1 . Remark 2.3. The requirement k > e + 1 is necessary: the existence of a Hadamard matrix of order 2A > 3 implies the existence of a partial A-geometry with k = 2 A and e = 2A - 1 (see [6, Proposition 2.3 and Lemma 1.81). Proof of Theorem 2.2. To begin, let {G, H } be a fixed pair of intersecting blocks of 3. Let ci be the number of blocks Y such that Gf Y f H , IG n YI >0, IH n Y1> 0 and 1 G n H n YI = i, 0 s i s A. Then the formula for c in Proposition 1.3 yields the equation
2 Ac,= k ( e + A - 1 ) - A ( e + l ) . A
t=O
Counting flags (x, Y) such that
XE
G n H and Hf Y f G, one obtains
A
1 ici = A(k -2).
(2)
i=O
Counting double flags (x, y, 2 ) such that x , y ~ G n H x, f y and G f Z f H , one gets
2 A
i=O
i(i-1)ci=A(A-1)(A-2).
(3)
We now multiply (1) by (j’+ j) where j is some integer, (2) by -2jA, and (3) by A, and add the three resultant equations. The left-hand side of this linear combination is 1 f(i)c, where
f(i) = i2A - i(2jA + A )
+ (j’A + jh).
Then f’(i) = 2iA -(2jA + A ) = 0 when i = j + f , and f”(i) = 2A > O ; so f(i) has an absolute minimum at i = j ++. Since f(j) = f ( j+ 1)= 0, f(i) 3 0 for all integral values of i. Since the ci’s are also all non-negative, the left-hand side of the linear combination of (1)-(3) is non-negative; and O S ( j ’ + j)[k(A + e - 1)- A(e + 1)]- 2jA(kA -2A)
+(A4-
3A3 .t 2A2).
Partial A-geometries of small nexus
23
then kj[2A2- ( j + l ) ( h+ e - 1 ) ] GA 4 - 3h3+ 2A2+4jh2- h ( e + l)(j”+j).
(4)
Then if 2A2- (j + l ) ( h+ e - 1 ) > 0 , we may replace k by ( e + 1) in (4), obtaining a strict inequality. Restating, we have
( e + 1)[2jh2- (j’
+ j ) ( A + e - l ) ]
-
hej” - hej - Aj” - A j
(5)
provided that 2A2-h-e+l >j . h+e-1 Now (5) simplifies to
O<(h4- 3h3+ 2h2)+j(-2eh2+ 2h2+e 2 - 1 ) + j 2 ( e 2 -1).
(7)
Consider the quantity d
+ 1).
=h2/(e
(8)
We wish to substitute [ d ] , the largest integer
(2A2- h - e + l ) ( e+ 1 ) > h2(h+ e - 1).
(9)
This inequality is equivalent to O>e2+e(-A2+A)+(A3-3h2+X - 1 ) - h ( e ) .
(10)
Now h(e)=O if and only if 2 e = h 2 - A + ( h 4 - 6 h 3 + 1 3 h 2 - 4 A + 4 ) 1 ’ 2 . Since (h2-3h+2)2
so g’(j) = 0 if and only if
2d-1 j = 2eh2-2h2-e2+1 -2e2- 2
2
.
Since g”(j) = 2e2- 2 > 0 , g ( j ) is a declining function to the left of j = i(2d - 1) and a rising function on the right. We wish to prove that g ( d - 1 ) = g ( d ) < O . It will follow that g ( [ d ] ) < O , violating (7) and proving the non-existence of a partial A-geometry with nexus e. One easily computes g ( d - 1 ) = g(d)+2eh2-2h2- e2+1 + ( 1 -2d)(e2- 1 ) = g ( d ) .
P.J. Cameron and D.A. Drake
24
Next, one computes
( e + l)g(d)/A2=2A2-3A(e+l ) + ( e + 1 ) 2 and concludes that g(d) <0 if A - 2 < e < 2A. We have proved
Proposition 2.4. There are no proper partial A-geometries whose parameters satisfy A > 3 . k > e + l and e<2A.
Then (6) is satisfied by j = 1 for all e <7. Setting A = 3 and < e < 7. Next, assume the existence of a partial A-geometry with parameter e = 2A. Then (7) simplifies to Now, assume A
= 3.
j = 1 in (7) yields a contradiction if A
0<(A4-3A3+2A2) +j(-4A3+6A2- 1)+j2(4A2-1).
(11)
The preceding inequality must hold when j satisfies (6); i.e., when
2A2-3A
+ 1> j(3A
1).
(12) Assume that A is an even integer. Then j=i(A-2) satisfies (12), hence may be substitued into (11) to yield O<-A2+2A, a contradiction. If A is odd and if e = 2A, one observes that j =;(A - 1) satisfies (12); then one may substitute j = ;(A - 1) into (11) to obtain the contradiction 0 < -A2 + 1. Lastly, assume that e = 2A + 1, A even. Then (7) becomes -
O<(A4-3A3+2A2)+ j(-4A3+4A2+4A)+ j2(4A2+4A). (13) The preceding inequality must hold when j satisfies (6); i.e., when 2A2- 3A > 3jA. Then (13) must hold when j =+(A - 2); this requirement yields another contradiction and thus completes the proof of Theorem 2.2. We conclude this section by considering the possible 3-geometries of nexus 7. Proposition 1.6 implies that 4k - 24 divides 2'' . 33 . 5 evenly. Then 4k - 24 = 2" . 3' . 5' where a < 10, b <3, c < 1. Since 4k - 24 is a multiple of 4, a 3 2. By Proposition 1.3, the integer d = ek/A =:k, so 3 I k. Then 4k -24 is a multiple of 3, so b 3 1. By Proposition 1.3, the integer u - k =&k(k - l ) ( k -3), so 7 I [k(k - l ) ( k -3)]. Then k '0, 1 or 3 (mod 7), and k = 2a-2. 3b . 5'+6, so
2"-2 * 3'
*
5' = 1, 2 or 4
(mod 7).
Thus 3b * 5" = 1, 2 or 4 (mod 7), and hence 3' . 5' is one of 9, 15, 135. We have proved that k = 6 + a . 2 @where a = 9 , 15 or 135 and O S p S 8 . Since A = 3 and e = 7, the integer in Proposition 1.5 is Dk =
k4-4k3 + 3 k 2 21(4k -24)
( k - 6)4+20(k - 6)3+ 147(k - 6)2+468(k - 6) + 540 84(k - 6) a 4 . 24p + 2oa3 . 23p + 1 4 7 .~ 220 ~ +46sa . 2 @+ 540 21a . 2 p + 2
Partial A-geometries of small nexus
25
+
If p 2 2, 2p 3 p 2; so 2p+2I 540, and hence p =s1. Then the only possible values of k are 15, 24, 21, 36, 141 and 276. One can see that Dk is not an integer for the two largest of these values, so we have proved
Proposition 2.5. If 3 is a partial 3-geometry with nexus 7, then the block size in 59 must be 7, 15, 21, 24 or 36. 3. A-quadrangles In this section, we define A-quadrangles. For A = 2, these are simply the proper partial A-geometries of nexus 3 discussed in Theorem 2.1. We will completely determine the A-quadrangles with A > 2.
Definition 3.1. A partial A-geometry will be called extremal if the condition lGnHHnKI>1 for blocks G, H, K implies that (GHnHflKIaA. Proposition 3.2. The inequality e 2 A* - A + 1 holds in every extremal partial A-geometry 3. Equality holds if and only i f the following criterion is satisfied: for each nonf7ag ( p , G), the e blocks through p which intersect G induce a projective plane of order A - 1 on the e points of G which are joined to p (degenerate unless A >2). Proof. Let ( p , G) be a nonflag, S be the set of e points of G which are joined to p . Since 3 is extremal, the blocks through p induce a “partial plane” E on S ; i.e., no two points of S lie in more than one common induced block. The number of pairs of points of S which are joined in E is ieA(A - 1) while the total number of pairs of points of S is + e ( e - I). Definition 3.3. A A-quadrangle is a proper extremal partial A-geometry with e = A * - A + I. We describe next a family of A-quadrangles. We refer to [l, 7, 81 for the geometric background: see in particular [l, Theorems 3.8 and 3.9 (Witt’s Theorem)].
Example 3.4. Let V be an %dimensional vector space over the field GF(q). Let Q be a quadratic form of Witt index 4 on V. The totally singular subspaces of dimension 4 will be called half spaces. The half spaces can be divided into two families with the property that, for any two half spaces S and T, the codimension of S n T in S is even if and only if S and T belong to the same family. (This is implicit in [8, 7.12 and 8.4.31; we give a short direct proof in the Appendix.) Define an incidence structure %(q)as follows: the points of %(q) are the totally
26
P.J. Cameron and D.A. Drake
singular 1-dimensional subspaces of V; the blocks are the members of one family of half spaces; and incidence is the relation of containment.
Proposition 3.5. % ( q ) is a ( q + 1)-quadrangle with e = q2 + q + 1, k q2+q+l.
= q3
+
Proof. Any block, with the points it contains, is a projective 3-space over GF(q); and two blocks meet in the empty set or a line. Given a totally singular 3-dimensional subspace U = ( u l , u2, u,), Witt’s Theorem yields a basis {ul,. . . , u4, w l , . . . , w4} for V consisting of singular vectors with B(u,,w , ) = 1, B(u,,u,) = B(u,, w,) = B(w,, w,) = 0 for i f j (where B is the bilinear form associated with 0).If follows that ( u l , u2, u,, u4) and ( u l , u2, ug, w4)are the only half spaces containing U and that exactly one of these is a block. Counting arguments (which we omit) now show that any point lies in 4 , + q2 + q + 1 blocks and that any two points on a singular line L lie in q + l blocks whose intersection is L. Of course, two points on a hyperbolic line cannot lie in a block. Finally, if S is a half space and p = ( u ) a point not in S, the map w -+ B(u, w ) is a nonzero linear form on S. A point incident with S is contained in the kernel of this map if and only if it is joined to p by a singular line. Then p is joined to q 2 + q + 1 points of S. It follows that the same number q 2 + q + 1 of blocks containing p meet S. This completes the proof. Our main result is the converse to Proposition 3.5
Theorem 3.6. A A-quadrangle with A 3 3 is isomorphic to %(q) where A
=q
+ 1.
Proof. We let % be a A -quadrangle with A 3 3. Put A = q + 1, hence e = q2+ q + 1. A non-empty intersection of two blocks will be called a line. If ( p , G) is a non-flag, the set n(p,G) of points of G collinear with p will be called a plane. Proposition 3.2 shows that a plane, equipped with the lines it contains, is a projective plane (of order 4 ) . It is clear that any two points of a block lie in a unique line, and we prove
Lemma 3.7. Three non-collinear points of a block lie in a unique plane. Proof. Let p , q, r be non-collinear points of G. There is a block H f G containing p and q ; r is joined to a point s of H outside G (indeed, to q2 such points); and p , q, r lie in the plane n(s,G). This plane is the smallest linear subspace containing p , q and r, hence is uniquely determined by these points. It follows from the classical axioms of projective geometry that a block, together with its points, lines and planes, is a projective space. We let d - 1 denote its dimension, so that k = (qd - l)/(q - 1).
Partial h-geometries of small nexus
21
Lemma 3.8. d is a multiple of 4.
Proof. Let G and H be disjoint blocks. Define a relation - on G by the rule p q if either p = q or there is a block containing p and q and meeting H. This relation is clearly reflexive and symmetric; we show it is transitive. Suppose p , q and r are distinct points with p q r. Let X and Y be blocks containing p , q and q, r, respectively, and meeting H. Then X n H and Y f l H are lines of the plane IT(q, H ) ; so there is a point s in X n Y n H. Then p, q, r lie in the plane n ( s , G); so there is a block containing p, r and s, whence p r. Thus is an equivalence relation. Clearly every equivalence class contains the line through any two of its points, hence is a subspace of the projective space G. The number of lines through a point p lying in the equivalence class is equal to the number of lines in the plane IT(p, H ) , that is, q 2 + q + 1; so the size of an equivalence class is q3+ q2+ q + 1. Since G is partitioned into equivalence classes, (q4- l)/(q - 1) divides (qd - l)/(q - l), whence 4 divides d .
-
- -
-
Lemma 3.9. d
-
= 4.
Proof. Proposition 1.6 asserts that k ( h + l)-A(e+ 1) divides- h 3 ( e+ l)’(Ae- 1) ( e -A). Substituting the values of A, e and k, we conclude that D(q, d) = (q + 2)(qd-4+qdp6+*
. .+ 1) - 1 divides
F ( q ) = ( q + 1)2(q2+q+2)2(q2+2q+2).
If q > 2 , each of (4’1)’ and (q2+q+2) is
-
Lemma 3.10. The geometry of points and lines is a polar space. Proof. We verify the axioms of Buekenhout and Shult [3]. It is clear that any line has at least three points and that no point is collinear with all others. Let L be a line and p a point not on L. We must show that p is collinear with one or all points of L. Let G be any block containing L. If P E G, then p is collinear with every point of L ; so suppose not. Then the points of L collinear with p are the
28
P.J. Cameron and D.A. Drake
members of cl(p, G) nL ; this set is a point or a line, since G is a projective 3-space. The rank of the polar space is four, since a block is a maximal totally singular subspace. Comparing the number of points of % (Lemma 1.2) with those in each of the six types of finite polar spaces of rank 4, we see that it is derived from a quadratic form of Witt index 4 on an 8-dimensional vector space; that is, it is of type D4(q)(see [8, p. 2191). Since the number of blocks is half the total number of half spaces and since two blocks meet in the empty set or a line, the blocks must comprise one family of half spaces. The theorem is proved.
Remark. Another way of making the identification is to observe that the only candidates for maximal totally singular subspaces containing a plane 17 = n(p,G) are the unique block containing 17 and union of the lines joining p to points of 17. Thus each plane lies in just two maximal subspaces. Now apply [8, 7.12 and 8.4.31.
Appendix Let V be a vector space of dimension 2 n over a commutative field, Q a quadratic form of Witt index n, €3 the associated bilinear form. As observed in the proof of Proposition 3.5, a totally singular subspace of dimension n - 1 lies in exactly two half spaces. Form a graph whose vertices are half spaces, two vertices adjacent if their intersection has dimension n - 1. Let S be a vertex, {TI,T2) an edge of the graph. Put T Ifl T2= U, and let d = dim(S n U ) .The functions s -+ B(s, u ) , u E U, form an ( n - d - 1)-dimensional subspace of the dual space S ’ ; its annihilator V is thus a (d + 1)-dimensional subspace containing S n U. Then (U, V) is a half space containing U, whence ( U , V) = TI or T2,say ( U , V) = T,. Then dim(S n T2)= dim(S f l T,)- 1. It follows that the distance function in the graph is given by the rule d ( S , T) = n -dim(S fl T), and that there are no circuits of odd length, that is, the graph is bipartite. Vertices in the same bipartite block lie at even distance. The claim made in Example 3.4 is proved.
Acknowledgment The second author wishes to thank Mark Hale for helpful discussions on symplectic geometries.
Partial A -geometries of small nexus
29
References [ l ] E. Artin, Geometric Algebra, Tracts in Pure and Applied Math. 3 (Interscience Publ., New York, 1957). [2] R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J . Math. 13 (1963) 389-419. [3] F. Buekenhout and E.E. Shult, On the foundations of polar geometry, Geometriae Dedicata 3 (1974) 155-170. [4] P.J. Cameron, Parallelisms of Complete Designs, London Math. SOC.,Lect. Note Series 23 (Cambridge Univ. Press, Cambridge, 1976). [S] P.J. Cameron and J.H. van Lint, Graph Theory, Coding Theory and Block Designs, London Math. SOC.,Lect. Note Series I9 (Cambridge Univ. Press, Cambridge, 1975). [6] D.A. Drake, Partial A -geometries and generalized Hadamard matrices over groups, to appear. [7] B. Huppert, Geometric algebra, Lecture notes, Dept. Math., Univ. of Illinois at Chicago Circle, Chicago, IL (1970). [XI J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 386 (Springer Verlag, Berlin, 1974).