arXiv subjects

Hendrik Van Maldeghem

Publications and source records attributed to Hendrik Van Maldeghem.

At least 37 records · Page 2Linked to original sources

arXiv2020

Moufang sets generated by translations in unitals

We consider unitals of order $q$ with two points which are centers of translation groups of order $q$. The group $G$ generated by these translations induces a Moufang set on the block joining the two points. We show that $G$ is either $\operatorname{SL}(2,\mathbb{F}_q)$ (as in all classical unitals and also in some non-classical examples), or $\operatorname{PSL}(2,\mathbb{F}_q)$, or a Suzuki or a Ree group. Moreover, $G$ is semi-regular outside the special block.

math.GR↗

arXiv2020

Automorphisms and opposition in spherical buildings of exceptional type, I

To each automorphism of a spherical building there is naturally associated an "opposition diagram", which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber then the automorphism is called "domestic". In this paper we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf{E}_6$, $\mathsf{F}_4$, and $\mathsf{G}_2$. Moreover, for all split spherical buildings of exceptional type we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most $2$ distinguished orbits encircled. Our results provide unexpected characterisations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

math.GR↗

arXiv2020

Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras

We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type $\widetilde{A}_2$ and respective absolute type $\widetilde{\mathsf{A}}_2$, $\widetilde{\mathsf{A}}_5$ and $\widetilde{\mathsf{E}}_6$.

math.CO↗

arXiv2020

Split buildings of type F4 in buildings of type E6

A symplectic polarity of a building Δ of type E6 is a polarity whose fixed point structure is a building of type F4 containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type F4). In this paper, we show in a geometric way that every building of type E6 contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type F4 fully embedded in the natural point-line geometry of Δ arises from a symplectic polarity.

math.CO↗

arXiv2020

Graphs, defined by Weyl distance or incidence, that determine a vector space

We study to which extent the family of pairs of subspaces of a vector space related to each other via intersection properties determines the vector space. In another language, we study to which extent the family of vertices of the building of a projective space related to each other via several natural respective conditions involving the Weyl distance and incidence determines the building. These results can be seen as generalizations of and variations on the Fundamental Theorem of Projective Geometry.

math.CO↗

arXiv2017

Distance regularity in buildings and structure constants in Hecke algebras

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.

math.CO↗

arXiv2016

On the varieties of the second row of the split Freudenthal-Tits Magic Square

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type $\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone's approach to finite quadric Veronesean varieties. This approach takes projective properties of complex Severi varieties as smooth varieties as axioms.

math.AG↗

arXiv2015

On the Lehmer Numbers, II

A composite number $n$ is called a Lehmer number when $ϕ(n) | n - 1$, where $ϕ$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $ϕ(n)| n-1$? 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.

math.NT↗

arXiv2015

Groups of Ree type in characteristic 3 acting on polytopes

Every Ree group $R(q)$, with $q\neq 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with $R(q) < G \leq \mathsf{Aut}(R(q))$ is not a C-group and therefore not the automorphism group of an abstract regular polytope of any rank.

math.GR↗

arXiv2014

The combinatorics of automorphisms and opposition in generalised polygons

We investigate the combinatorial interplay between automorphisms and opposition in (primarily finite) generalised polygons. We provide restrictions on the fixed element structures of automorphisms of a generalised polygon mapping no chamber to an opposite chamber. Furthermore, we give a complete classification of automorphisms of finite generalised polygons which map at least one point and at least one line to an opposite, but map no chamber to an opposite chamber. Finally, we show that no automorphism of a finite thick generalised polygon maps all chambers to opposite chambers, except possibly in the case of generalised quadrangles with coprime parameters.

math.CO↗

arXiv2013

Imbrex geometries

We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a Lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety $\mathcal{E}_{6,1}$ by means of a local condition on tangent spaces.

math.MG↗

arXiv2012

Descent of affine buildings - I. Large minimal angles

In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a (skew) field with a complete valuation, as proposed by Jacques Tits. This first part lays the foundations for our approach and deals with the `large minimal angle' case.

math.MG↗

arXiv2012

Projective planes over quadratic 2-dimensional algebras

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry A2 \times A2 in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization of "projective planes over quadratic 2-dimensional algebras", in casu the split and non-split Galois extensions and the dual numbers.

math.AG↗

arXiv2012

Automorphisms and opposition in twin buildings

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore we show that no automorphism of a locally finite 2-spherical twin building of rank at least 3 maps every residue of one fixed type to an opposite. The main ingredient of the proof is a lemma that states that every duality of a thick finite projective plane admits an absolute point, i.e., a point mapped onto an incident line. Our results also hold for all finite irreducible spherical buildings of rank at least 3, and as a consequence we deduce that every involution of a thick irreducible finite spherical building of rank at least 3 has a fixed residue.

math.CO↗

arXiv2011

A classification of commutative parabolic Hecke algebras

Let $(W,S)$ be a Coxeter system with $I\subseteq S$ such that the parabolic subgroup $W_I$ is finite. Associated to this data there is a \textit{Hecke algebra} $\scH$ and a \textit{parabolic Hecke algebra} $\scH^I=\mathbf{1}_I\scH\mathbf{1}_I$ (over a ring $\ZZ[q_s]_{s\in S}$). We give a complete classification of the commutative parabolic Hecke algebras across all Coxeter types.

math.RT↗

arXiv2011

Compact Totally Disconnected Moufang Buildings

Let $Δ$ be a spherical building each of whose irreducible components is infinite, has rank at least 2 and satisfies the Moufang condition. We show that $Δ$ can be given the structure of a topological building that is compact and totally disconnected precisely when $Δ$ is the building at infinity of a locally finite affine building.

math.GR↗

arXiv2011

Codistances of 3-spherical buildings

We show that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning (i.e., is one half of a twin building) as soon as it admits a codistance, i.e., a twinning with a single chamber.

math.GR↗

arXiv2009

Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings

In this paper, we show that the building at infinity of a two-dimensional affine R-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type G~_2, and it reproves the results obtained mainly by the second author for types A~_2 and C~_2. The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional R-building).

math.MG↗