Maps related to polar spaces preserving a Weyl distance or an incidence condition

Abstract

Let Ω i and Ω j be the sets of elements of respective types i and j of a polar space ∆ of rank at least 3, viewed as a Tits-building. For any Weyl distance δ between Ω i and Ω j , we show that δ is characterised by i and j and two additional numerical parameters k and . We consider permutations ρ of Ω i ∪ Ω j that preserve a single Weyl distance δ. Up to a minor technical condition on , we prove that, up to trivial cases and two classes of true exceptions, ρ is induced by an automorphism of the Tits-building associated to ∆, which is always a type-preserving automorphism of ∆ (and hence preserving all Weyl-distances), unless ∆ is hyperbolic, in which case there are outer automorphisms. For each class of exceptions, we determine a Tits-building ∆ in which ∆ naturally embeds and is such that ρ is induced by an automorphism of ∆ . At the same time, we prove similar results for permutations preserving a natural incidence condition. These yield combinatorial characterisations of all groups of algebraic origin which are the full automorphism group of some polar space as the automorphism group of many bipartite graphs.(PS3) For U ∈ Ω with dim(U) = n − 1 and p ∈ X \ U, the union of all elements of Ω of dimension 1 containing p and intersecting U nontrivially is an element of Ω of dimension n − 1 which intersects U in a hyperplane.(PS4) There are two disjoint elements of Ω of dimension n − 1.A set X of cardinality at least two, together with Ω = X ∪ { } is considered to be a polar space of rank 1. Henceforth, ∆ denotes a polar space of rank n with n ≥ 2.Collinearity and opposition − An element of Ω of dimension n − 1 is called a maximal singular subspace (MSS for short) and an element of Ω of dimension 1 is called a line. Let x and y be two distinct points. If they are on a common line, they are called collinear and we write x ⊥ y, if not, they are called opposite. The set of points equal or collinear with x is denoted by x ⊥ . A subspace S of ∆ is a subset of X such that the lines joining any two collinear points of S are contained in S. Moreover, if S contains no pair of opposite points, the subspace is called singular. The elements of Ω are precisely the singular subspaces of ∆. If U and V are singular subspaces with U ⊆ V , then the codimensionFor a singular subspace U, we define U ⊥ as x∈U x ⊥ . For any singular subspace V , we say that U and V are collinear if V ⊆ U ⊥ . If they are collinear but disjoint, we write U ⊥ V . Let T be a set of pairwise collinear singular subspaces. We denote by 〈T 〉 the smallest singular subspace containing all members of T , and we also say that the members of T generate 〈T 〉 or that 〈T 〉 is spanned by the members of T . If T consists of two distinct collinear points x, y, we denote the unique line joining these points by xy. The projection proj V (U) of a singular subspace U on a singular subspace V is V ∩ U ⊥ and the subspace spanned by U and proj V (U) is denoted byis empty, we say that U and V are semi-opposite. Now let U and V be semi-opposite singular subspaces. If dim(U) = dim(V...

Computations regarding certain graphs associated to finite polar spaces

“…De Schepper and Van Maldeghem [3] answer Problem 1.1 in the affirmative but for the case where i < ℓ = n − 1, which cannot be treated by the techinque they exploit in [3]. A subcase of this wild case is studied by De Schepper and the author in [2]. Explicitly, assuming that i + j − k = ℓ = n − 1, an affirmative answer is obtained for Problem 1.1.…”

“…Explicitly, assuming that i + j − k = ℓ = n − 1, an affirmative answer is obtained for Problem 1.1. In particular it is proved in [2] that, when i + j − k = n − 1 = ℓ, Problem 1.1 can be answered in the affirmative provided that the following is true:…”

“…Theorem 1.2 is proved in [2] by purely geometrical arguments, valid when the lines of ∆ have at least four points. When the lines of ∆ have size 3 a different approach is necessary, relying on computations.…”

“…In this paper we shall perform those computations, in a more general setting than [2] would require. Indeed we only assume finiteness while in view of [2] we could restrict ourselves to the case where the lines of ∆ have just three points.…”

“…In this paper we shall perform those computations, in a more general setting than [2] would require. Indeed we only assume finiteness while in view of [2] we could restrict ourselves to the case where the lines of ∆ have just three points. However, treating the general case, with lines of size s + 1 instead of 3, makes no difference: it just amounts to write s instead of 2.…”