. Here the flats of H(2d 1, q 2 ) consist of all nonzero totally isotropic

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1 NEW BOUNDS FOR PARTIAL SPREADS OF H(d 1, ) AND PARTIAL OVOIDS OF THE REE-TITS OCTAGON FERDINAND IHRINGER, PETER SIN, QING XIANG ( ) Abstract Our first result is that the size of a partial spread of H(, ) is at most p +p t+1, where = p t, p is a prime For fixed p this bound is in o( ), which is asymptotically better than the previous best known bound of ( + + )/ We give similar bounds for partial spreads of H(d 1, ), d even Our second result is that the size of a partial ovoid of the Ree-Tits octagon O( t ) is at most 6 t + 1 This bound, in particular, implies that the Ree-Tits octagon O( t ) does not have an ovoid 1 Introduction Determining cliue numbers of graphs is a traditional topic of combinatorics Partial spreads of polar spaces are cliues of the oppositeness graph defined on the generators of the polar space While bounds for partial spreads and partial ovoids are a long studied topic since Thas popularized the problem in [10], many uestions remain open The previous best known bound on the size of partial spreads of H(d 1, ), d even and = p t (p a prime), is in Θ( d 1 ) [6, 7], while even for d = the best known examples only have size Θ( ) [4] Here we provide the first upper bound which is in o( d 1 ) for fixed p We note that the size of the largest partial spreads in H(d 1, ), d odd, is known to be d + 1 [1, 1] The only known thick finite generalized octagons are the Ree-Tits octagons O( t ), t odd, and their duals, which were defined in [11] For O(), it was already shown [5] that a partial ovoid has at most 7 points, while for the t cases even the existence of ovoids, which are partial ovoids of size 64 t + 1, was an open uestion Coolsaet and Van Maldeghem [5, p 108] conjectured that the Ree-Tits octagon O( t ) does not have an ovoid We prove this conjecture by giving the upper bound 6 t + 1 on the size of a partial ovoid of O( t ), t odd This nonexistence result further implies that the Ree-Tits octagon does not admit a line-domestic collineation [8, p 58] A Hermitian polar space H(d 1, ) is the incidence geometry arising from a non-degenerate Hermitian form f of F d Here the flats of H(d 1, ) consist of all nonzero totally isotropic subspaces of F d with respect to the form f; incidence is the inclusion relation of the flats of H(d 1, ) The maximal totally isotropic subspaces of F d with respect to the form f are called generators of H(d 1, ) A partial spread of H(d 1, ) is a set of pairwise disjoint generators of H(d 1, ) A simple double counting argument shows that a partial spread of H(d 1, ) has size at most d Our first result is the following Theorem 11 Let = p t with p prime and t 1 Let Y be a partial spread of H(d 1, ), where d is even ( ) t p (a) If d =, then Y +p + 1 (b) If d = and p =, then Y 19 t Key words and phrases Association scheme, generalized n-gon, Hermitian polar space, oppositeness graph, ovoid, partial ovoid, p-rank, partial spread, spread, the Ree-Tits octagon This work was partially supported by a grant from the Simons Foundation (#04181 to Peter Sin) 1

2 IHRINGER, SIN, AND XIANG ( ) t (c) If d >, then Y p d 1 p pd 1 p For fixed p these bounds are, unlike all other known bounds, asymptotically better than o( d 1 ) For d = the previous best known bound is ( + + )/ [6] An easy calculation shows that this old bound is better than the bound in part (a) of Theorem 11 if p = and t or if t = 1 But for fixed p (and let = p t ), the bound in part (a) of Theorem 11 is in o( ), which is asymptotically better than the bound of ( + + )/ For d > the new bound improves all previous bounds if t > 1 [6, 7] A generalized n-gon of order (s, r) is a triple Γ = (P, L, I), where elements of P are called points, elements of L are called lines, and I P L is an incidence relation between the points and lines, which satisfies the following axioms [1]: (a) Each line is incident with s + 1 points (b) Each point is incident with r + 1 lines (c) The incidence graph has diameter n and girth n Here the incidence graph is the bipartite graph with P L as vertices, p P and l L are adjacent if (p, l) I A partial ovoid of a generalized n-gon Γ is a set of points pairwise at distance n in the incidence graph An easy counting argument shows that the size of a partial ovoid of a generalized octagon of order (s, r) is at most (sr) +1 A partial ovoid of a generalized octagon of order (s, r) is called an ovoid if it has the maximum possible size (sr) + 1 The Ree-Tits octagon O( t ) is a generalized octagon of order ( t, 4 t ), so the size of a partial ovoid is at most 64 t + 1 Our second result is the following Theorem 1 The size of a partial ovoid of the Ree-Tits octagon O( t ), t odd, is at most 6 t + 1 This extends a result by Coolsaet and Van Maldeghem [5], who established an upper bound of 7 for O() Theorem 1 implies that the Ree-Tits octagon O( t ) does not have an ovoid A computer search showed that for O() the largest partial ovoid has size 4, so Theorem 1 is not tight Two Association Schemes We need some basic properties of an association scheme defined on the generators of H(d 1, ) A complete introduction to association schemes can be found in [, Ch ] Definition 1 Let X be a finite set of size n An association scheme with d + 1 classes is a pair (X, R), where R = {R 0,, R d } is a set of symmetric binary relations on X with the following properties: (a) R is a partition of X X (b) R 0 is the identity relation (c) There are numbers p k ij such that for x, y X with xr k y there are exactly p k ij elements z with xr i z and zr j y The relations R i are described by their adjacency matrices A i C n,n defined by { 1, if xr i y, (A i ) xy = 0, otherwise

3 PARTIAL SPREADS AND PARTIAL OVOIDS In this paper the matrix A d is referred to as the oppositeness matrix Denote the all-ones matrix by J There exist (see [, p 45]) idempotent Hermitian matrices E j C n,n with the properties E j = I, E 0 = n 1 J, j=0 A j = P ij E i, i=0 E j = 1 n Q ij A i, where P = (P ij ) C d+1,d+1 and Q = (Q ij ) C d+1,d+1 are the so-called eigenmatrices of the association scheme The P ij are the eigenvalues of A j The multiplicity f i of P ij satisfies f i = rank(e i ) = tr(e i ) = Q 0i In this paper we consider the association schemes corresponding to the dual polar graph of H(d 1, ) and the Ree-Tits octagon For the dual polar graph of H(d 1, ) we have the following situation Here X is the set of generators of H(d 1, ) and two generators a, b are in relation R i, where 0 i d, if and only if a and b intersect in codimension i The dual polar graph of H(d 1, ) has diameter d and is distance regular As in [7], for the dual polar graph of H(d 1, ) we obtain and f d = d 1 d i=0 = d 1 d 1 + 1, Q id = P di P 0i Q 0d = f d ( ) i For the Ree-Tits octagon O( t ) we consider the following association scheme: the set X is the set of points of the octagon, and two points a and b are in relation R i, where 0 i 4, if their distance in the incidence graph of O( t ) is i Notice that A 4 is the oppositeness matrix in this case We will use the following observation p-ranks of the Oppositeness Matrices Lemma 1 Let A and B be n n integer matrices and p be a prime If A B mod p, then rank p (A) rank(b) Proof We have rank p (A) = rank p (B) rank(b) Lemma (a) The p-rank of the oppositeness matrix of lines of H(, p ) is p +p (b) The p-rank of the oppositeness matrix of generators of H(d 1, p ), d even, is at most p d 1 p pd 1 p+1 Proof Notice that H(, p ) is dual to Q (5, p) By [9, Example 6], the p-rank of the oppositeness matrix A is eual to the dimension of L((p 1)ω 1 )This value was calculated in [, Theorem 1] Applying Theorem 1 of [] with l = and r = p 1, for the oppositeness matrix A we obtain Part (a) of the lemma follows rank p (A ) = p + p

4 4 IHRINGER, SIN, AND XIANG For (b) notice that the matrix E d of the dual polar graph of H(d 1, p ) has rank f d = p d 1 p pd 1 p + 1 The matrix np d E d has only integer entries and we have A d np d 1 E d mod p for d even By Lemma 1, rank p (A d ) rank(e d ) = p d 1 p pd 1 p+1 Proposition (Sin [9, Prop 5]) Let A R () denote the oppositeness matrix for objects of one fixed type in a building with root system R over F, where = p t, p is a prime Then rank p (A R ()) = rank p (A R (p)) t According to [9, p 8, para ], the -rank of the oppositeness matrix of O() corresponds to the dimension of a simple module of the group F 4 (ω 4 in the notation of [9]) Veldkamp calculated all simple modules of the group F 4 in characteristic [15, Table ] As for this particular module it is the modulo reduction of the corresponding Weyl module in characteristic zero, whose dimension was known before It is also easy to compute the oppositeness matrix of O() and its -rank by computer Hence, we obtain the following result Lemma 4 The -rank of the oppositeness matrix of O() is 6 4 Proofs of the Results Our main result will follow from the following lemma Lemma 41 Let (X, ) be a graph Let A be the adjacency matrix of X Let Y be a cliue of X Then { rank p (A) + 1, if p divides Y 1, Y rank p (A), otherwise Proof Let J be the all-ones matrix of size Y Y Let I be the identity matrix of size Y Y As Y is a cliue, the submatrix A of A indexed by Y is J I Hence, the submatrix has p-rank Y 1 if p divides Y 1, and it has p-rank Y if p does not divide Y 1 As rank p (A ) rank p (A), the assertion follows Proof of Theorem 11 and Theorem 1 Combining Lemma, Proposition and Lemma 41 shows Theorem 11 Notice here that p +p is divisible by p unless p = Combining Proposition, Lemma 4 and Lemma 41 shows Theorem 1 5 Final Remarks The used techniue also works for partial ovoids of the twisted triality hexagon of order (, ), where we obtain the p-rank ((4p 5 + p)/5) t, but there a better bound of order + 1 is known [14, Theorem 6419] For all other known generalized polygons p-ranks do not improve any known bounds for partial ovoids The bounds on partial spreads H(d 1, ), d > even, in Theorem 11 can be improved by calculating exact p-ranks We expect these bounds to be much better than the given ones as for H(5, ) the corresponding p-rank is ((11p 5 + 5p + 4p)/0) t, while an argument as in Lemma (b) only yields (p 5 p 4 + p p + p) t as an upper bound Unfortunately, calculating the exact p-rank seems to be too complicated Notice that this is a finite problem for fixed p and fixed d as one only has to calculate the p-rank for t = 1 by Proposition

5 PARTIAL SPREADS AND PARTIAL OVOIDS 5 Acknowledgments The first author thanks John Bamberg and Klaus Metsch for their remarks References [1] A Aguglia, A Cossidente, and G L Ebert Complete spans on Hermitian varieties In Proceedings of the Conference on Finite Geometries (Oberwolfach, 001), volume 9, pages 7 15, 00 [] O Arslan and P Sin Some simple modules for classical groups and p-ranks of orthogonal and Hermitian geometries J Algebra, 7: , 011 [] A E Brouwer, A M Cohen, and A Neumaier Distance-regular graphs, volume 18 of Ergebnisse der Mathematik und ihrer Grenzgebiete () [Results in Mathematics and Related Areas ()] Springer-Verlag, Berlin, 1989 [4] K Coolsaet Some large partial ovoids of Q (5, ), for odd Des Codes Cryptogr, 7(1):119 18, 014 [5] K Coolsaet and H Van Maldeghem Some new upper bounds for the size of partial ovoids in slim generalized polygons and generalized hexagons of order (s, s ) J Algebraic Combin, 1():107 11, 000 [6] J De Beule, A Klein, K Metsch, and L Storme Partial ovoids and partial spreads in Hermitian polar spaces Des Codes Cryptogr, 47(1-):1 4, 008 [7] F Ihringer A new upper bound for constant distance codes of generators on hermitian polar spaces of type H(d 1, ) J Geom, 105(): , 014 [8] J Parkinson, B Temmermans, and H Van Maldeghem The combinatorics of automorphisms and opposition in generalised polygons Ann Comb, 19(): , 015 [9] P Sin Oppositeness in buildings and simple modules for finite groups of Lie type In Buildings, finite geometries and groups, volume 10 of Springer Proc Math, pages 7 86 Springer, New York, 01 [10] J A Thas Ovoids and spreads of finite classical polar spaces Geom Dedicata, 10(1-4):15 14, 1981 [11] J Tits Moufang octagons and the Ree groups of type F 4 Amer J Math, 105():59 594, 198 [1] H Van Maldeghem Generalized Polygons Birkhäuser Basel, 1998 [1] F Vanhove The maximum size of a partial spread in H(4n + 1, ) is n Electron J Combin, 16(1):Note 1, 6, 009 [14] F Vanhove Incidence geometry from an algebraic graph theory point of view PhD thesis, Ghent University, 011 [15] F D Veldkamp Representations of algebraic groups of type F 4 in characteristic J Algebra, 16:6 9, 1970 Ferdinand Ihringer, Mathematisches Institut, Justus-Liebig-Universität, D-59 Giessen, Germany address: FerdinandIhringer@mathuni-giessende Peter Sin, Department of Mathematics, University of Florida, Gainesville, FL 611, USA address: sin@ufledu Qing Xiang, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA address: xiang@udeledu