Papers by Hendrik Van Maldeghem
Regularity, Antiregularity and 3-Regularity
Translation Generalized Quadrangles, 2006
Springer Proceedings in Mathematics & Statistics, 2014
A triality of type I id in a building of type D 4 is a type rotating automorphism of order 3 whos... more A triality of type I id in a building of type D 4 is a type rotating automorphism of order 3 whose structure of fixed flags is the building of type G 2 related to Dickson's simple groups (in geometric term, this building is the split Cayley generalized hexagon over the field in question). Such a triality exists over any field and is unique up to conjugacy. In this paper, we present two characterizations of such trialities among all type rotating automorphisms (hence not necessarily of order 3). We prove that, if for a type rotating automorphism ✓ of , no non-fixed line and its image are contained in adjacent chambers, and ✓ fixes at least one line, then ✓ is a triality of type I id (here, lines are vertices of type 2, with Bourbaki labeling). Also, if a type rotating automorphism ✓ of never maps a line to an opposite line, then it is also a triality of type I id . We moreover show that this condition is equivalent with ✓ not mapping any chamber to an opposite one. The latter completes the programme for type rotating automorphisms of buildings of type D 4 of determining all domestic automorphisms of spherical buildings.
Asymptotic normality of the generalized Eulerian numbers
Ars Combinatoria, 1998
arXiv (Cornell University), 2015
A composite number $n$ is called a Lehmer number when $\phi(n) | n - 1$, where $\phi$ is the Eule... more A composite number $n$ is called a Lehmer number when $\phi(n) | n - 1$, where $\phi$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $\phi(n)| n-1$? No such numbers are known. In this paper we establish an Euler Totient Inequality and relate some new parameters to Lehmer numbers. As an application of what we have done, we show that for all prime numbers $p$ and for all odd square-free numbers $n$, at most one of $n$ or $pn$ is a Lehmer number. Finally we suggest some open problems for the future investigations on the Lehmer numbers.
Common characterizations of the finite Moufang polygons
Oxford University Press eBooks, Aug 21, 1991
Jacques Tits, Œuvres – Collected Works
Jacques Tits, Œuvres – Collected Works
Collection de quatre volumes reproduisant l'essentiel des oeuvres mathematiques de Jacques Tits
Israel Journal of Mathematics, Dec 1, 1999
THEOREM A: If ~3 is an infinite Moufang polygon of finite Morley rank, then ~3 is either the proj... more THEOREM A: If ~3 is an infinite Moufang polygon of finite Morley rank, then ~3 is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ~3 is an algebraic polygon. It follows that any infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank 2 is either PSL3(K), PSp4(K ) or G2(K) for some algebraically closed field K. Spherical irreducible buildings of Tits rank _> 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem A we show THEOREM B: If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ~ 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraically closed field.
Contributions to Discrete Mathematics, Dec 10, 2009
In this paper, we construct the Hall-Janko graph within the split Cayley hexagon H(4). Using this... more In this paper, we construct the Hall-Janko graph within the split Cayley hexagon H(4). Using this graph we then construct the near-octagon of order (2, 4) as a subgeometry of the dual of H(4), with J2 : 2 as its automorphism group. These constructions are based on a lemma determining the possibilities for the structure of the intersection of two subhexagons of order 2 in H(4).
Innovations in incidence geometry, 2008
In this paper we continue our study begun in aiming at characterizing the embedding of the split ... more In this paper we continue our study begun in aiming at characterizing the embedding of the split Cayley hexagons H(q), q even, in PG(5, q) by intersection numbers with respect to their lines. We prove that, for q = 3, every pseudo-hexagon (i.e. a set L of lines of PG(5, q) with the properties that (1) every plane contains 0, 1 or q + 1 elements of L, (2) every solid contains no more than q 2 + q + 1 and no less than q + 1 elements of L, and (3) every point of PG(5, q) is on q + 1 members of L) which is 1-polarized at some point x (i.e., the lines of L through x do not span PG(5, q)) is either the line set of the standard embedding of H(q) in PG(5, q), or q = 2 (in the latter case all pseudo-hexagons are classified in ).
Characterizations of groups by geometries and geometries by groups
Non UBCUnreviewedAuthor affiliation: Ghent UniversityFacult
Two Characterizations of the Hermitian Spread in the Split Cayley Hexagon
Springer eBooks, 2001
From the Hermitian spread in the generalized hexagon H (q), we construct a certain geometry Γ S ,... more From the Hermitian spread in the generalized hexagon H (q), we construct a certain geometry Γ S , which is a generalized quadrangle. The fact that Γ S is a generalized quadrangle turns out to characterize the Hermitian spread as a spread of H (q). Furthermore, we give a characterization of this spread using the group of projectivities induced by the spread lines.
Discrete Mathematics, Sep 1, 2003
We generalize the notion of a semi-a ne plane to structures with higher girth n. We prove that, i... more We generalize the notion of a semi-a ne plane to structures with higher girth n. We prove that, in the ÿnite case, for n odd, and with an additional assumption also for n even, these geometries, which we call forgetful n-gons, always arise from (ÿnite) generalized n-gons by 'forgetting' lines.
Ovoids, Spreads and Self-Dual Polygons
Ovoids and spreads are in more than one respect special configurations in generalized polygons. R... more Ovoids and spreads are in more than one respect special configurations in generalized polygons. Roughly speaking, an ovoid in a generalized polygon is a set of mutually opposite points “of maximal size” (this will be made precise below for infinite polygons), and the dual notion is a spread. Sets of maxial size with respect to a certain property in geometries usually themselves have interesting properties. For example, they might be used to construct other geometries (as ovoids in projective spaces are used to construct generalized quadrangles; see Subsection 3.7.1 on page 120. By the way, the term “ovoid” might seem confusing here; one connection is that in a lot of cases, ovoids of W(K), with K of characteristic 2, are via the standard embedding of W(K) in PG(3, K) ovoids of PG(3, K); see Proposition 7.6.14 and Subsection 7.6.25 below. Another connection is that polarities of projective spaces sometimes produce ovoids in these spaces, and likewise polarities in generalized polygons sometimes produce ovoids in the polygons). Another feature of ovoids is that they sometimes have interesting automorphism groups. The Suzuki groups and the Ree groups of characteristic 2 arise in that way. We will use the ovoids to prove some properties of these groups.
Electronic Journal of Combinatorics, Feb 4, 2009
In this paper we improve on a result of Beutelspacher, De Vito & Lo Re, who characterized in 1995... more In this paper we improve on a result of Beutelspacher, De Vito & Lo Re, who characterized in 1995 finite semiaffine spaces by means of transversals and a condition on weak parallelism. Basically, we show that one can delete that condition completely. Moreover, we extend the result to the infinite case, showing that every plane of a planar space with at least two planes and such that all planes are semiaffine, comes from a (Desarguesian) projective plane by deleting either a line and all of its points, a line and all but one of its points, a point, or nothing.
Designs, Codes and Cryptography, May 1, 2004
It is shown that if a spread of a finite split Cayley hexagon is translation with respect to two ... more It is shown that if a spread of a finite split Cayley hexagon is translation with respect to two disjoint flags then it is either a hermitian spread or a Ree-Tits spread. Analogously, if an ovoid of a classical generalized quadrangle Qð4; qÞ is translation with respect to two disjoint flags then it is either an elliptic quadric or a Suzuki-Tits ovoid. In the course of obtaining these results, we introduce the notion of local polarity for ovoid-spread pairings and show that if an ovoid-spread pairing is locally polar at each of its elements then it arises from a polarity.
European Journal of Combinatorics, Nov 1, 1993
Archiv der Mathematik, Oct 1, 2003
We show that, if a collineation group G of a generalized (2n + 1)-gon has the property that every... more We show that, if a collineation group G of a generalized (2n + 1)-gon has the property that every symmetry of any apartment extends uniquely to a collineation, then is the unique projective plane with 3 points per line (the Fano plane) and G is its full collineation group. A similar result holds if one substitutes "apartment" with "path of length 2k 2n + 2".
Glasgow Mathematical Journal, Sep 1, 1997
We show in a direct and elementary way that the spherical building at infinity of every rank 3 af... more We show in a direct and elementary way that the spherical building at infinity of every rank 3 affine building which satisfies Tits' Moufang condition, is itself a Moufang building. This result is also true for higher rank affine buildings by Tits' classification . The Moufang condition for-not necessarily spherical-buildings was introduced by Tits [5]. It generalizes the usual Moufang condition for spherical buildings, see Tits [3], which on its turn was a generalization of the Moufang condition for projective planes. The Moufang condition seems to be the most natural condition under which a classification of certain classes of buildings is possible. For spherical buildings of rank s 3, and for affine building of rank s 4, this is trivially true for all those buildings are classified without any supplementary condition. For spherical buildings of rank 2, Tits [3] announces such classification and partial results have been published. There seems to be no further explicit classification of Moufang affine buildings of rank 3 in the literature. In this short note, we show that such a classification can be reduced to checking the Moufang property in the "known classical buildings". Our method uses the building at infinity of the affine building. The definition of Moufang affine building does not imply ipso facto that the building at infinity also satisfies the Moufang condition. We will show that this is however a consequence, using elementary techniques. So our main result reads: MAIN RESULT. The building at infinity of an irreducible Moufang rank 3 affine building is a Moufang rank 2 spherical building.
Journal of Algebra, Jul 1, 2017
In this paper we define generalised spheres in buildings using the simplicial structure and Weyl ... more In this paper we define generalised spheres in buildings using the simplicial structure and Weyl distance in the building, and we derive an explicit formula for the cardinality of these spheres. We prove a generalised notion of distance regularity in buildings, and develop a combinatorial formula for the cardinalities of intersections of generalised spheres. Motivated by the classical study of algebras associated to distance regular graphs we investigate the algebras and modules of Hecke operators arising from our generalised distance regularity, and prove isomorphisms between these algebras and more well known parabolic Hecke algebras. We conclude with applications of our main results to non-negativity of structure constants in parabolic Hecke algebras, commutativity of algebras of Hecke operators, double coset combinatorics in groups with BN -pairs, and random walks on the simplices of buildings.
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Papers by Hendrik Van Maldeghem