The Valuations of the Near Polygon Gn

We show that every valuation of the near 2n-gon Gn, n > 2, is induced by a unique classical valuation of the dual polar space DH(2n − 1, 4) into which Gn is isometrically embeddable. 1 Basic definitions and main results A near polygon is a connected partial linear space S = (P,L, I), I ⊆ P × L, with the property that for every point x and every line L, there exists a unique point on L nearest to x. Here, distances d(·, ·) are measured in the collinearity graph Γ of S. If d is the diameter of Γ, then the near polygon is called a near 2d-gon. A near 0-gon is a point and a near 2-gon is a line. Near quadrangles are usually called generalized quadrangles. If X1 and X2 are two nonempty sets of points of S, then d(X1, X2) denotes the smallest distance between a point of X1 and a point of X2. If X1 is a singleton {x}, then we will also write d(x,X2) instead of d({x}, X2). For every i ∈ N and every nonempty set X of points of S, Γi(X) denotes the set of all points x ∈ X for which d(x,X) = i. If X is a singleton {x}, then we will also write Γi(x) instead of Γi({x}). Let S be a near polygon. A set X of points of S is called a subspace if every line of S having two of its points in X has all its points in X. If X is a subspace, then we denote by X̃ the subgeometry of S induced on the point set X by those lines of S which have all their points in X. A set X of points of S is called convex if every point on a shortest path between two points of X is also contained in X. If X is a non-empty convex subspace of S, then X̃ is also a near polygon. Clearly, the intersection of any number of (convex) subspaces is again a (convex) subspace. If ∗1, ∗2, . . . , ∗k are k > 1 objects (i.e., points or nonempty sets of points) of S, then 〈∗1, ∗2, . . . , ∗k〉 denotes the smallest convex subspace ∗Postdoctoral Fellow of the Research Foundation Flanders the electronic journal of combinatorics 16 (2009), #R137 1 of S containing ∗1, ∗2, . . . , ∗k. The set 〈∗1, ∗2, . . . , ∗k〉 is well-defined since it equals the intersection of all convex subspaces containing ∗1, ∗2, . . . , ∗k. A near polygon S is called dense if every line is incident with at least three points and if every two points at distance 2 have at least two common neighbors. If x and y are two points of a dense near polygon S at distance δ from each other, then by Brouwer and Wilbrink [6, Theorem 4], 〈x, y〉 is the unique convex subspace of diameter δ containing x and y. The convex subspace 〈x, y〉 is called a quad if δ = 2, a hex if δ = 3 and a max if δ = n− 1. We will now describe two classes of dense near polygons. (I) Let n > 2, let K be a field with involutory automorphism ψ and let K denote the fix field of ψ. Let V be a 2n-dimensional vector space over K equipped with a nondegenerate skew-ψ-Hermitian form fV of maximal Witt index n. The subspaces of V which are totally isotropic with respect to fV define a Hermitian polar space H(2n − 1,K /K). We denote the corresponding Hermitian dual polar space by DH(2n − 1,K/K). So, DH(2n− 1,K/K) is the point-line geometry whose points, respectively lines, are the ndimensional, respectively (n− 1)-dimensional, subspaces of V which are totally isotropic with respect to fV , with incidence being reverse containment. The dual polar space DH(2n − 1,K/K) is a dense near 2n-gon. In the finite case, we have K ∼= Fq and K ∼= Fq2 for some prime power q. In this case, we will denote DH(2n − 1,K /K) also by DH(2n − 1, q). The dual polar space DH(3, q) is isomorphic to the generalized quadrangle Q(5, q) described in Payne and Thas [24, Section 3.1]. (II) Let n > 2, let V be a 2n-dimensional vector space over F4 with basis B = {ē1, ē2, . . . , ē2n}. The support of a vector x̄ = ∑2n i=1 λiēi of V is the set of all i ∈ {1, . . . , 2n} satisfying λi 6= 0; the cardinality of the support of x̄ is called the weight of x̄. Now, we can define the following point-line geometry Gn(V,B). The points of Gn(V,B) are the ndimensional subspaces of V which are generated by n vectors of weight 2 whose supports are two by two disjoint. The lines of Gn(V,B) are of two types: (a) Special lines: these are (n−1)-dimensional subspaces of V which are generated by n− 1 vectors of weight 2 whose supports are two by two disjoint. (b) Ordinary lines: these are (n− 1)-dimensional subspaces of V which are generated by n−2 vectors of weight 2 and 1 vector of weight 4 such that the n−1 supports associated with these vectors are mutually disjoint. Incidence is reverse containment. By De Bruyn [10] (see also [11, Section 6.3]), the geometry Gn(V,B) is a dense near 2n-gon with three points on each line. The isomorphism class of the geometry Gn(V,B) is independent from the vector space V and the basis B of V . We will denote by Gn any suitable element of this isomorphism class. The near polygon G2 is isomorphic to the generalized quadrangle Q (5, 2). Now, endow the vector space V with the (skew-)Hermitian form fV which is linear in the first argument, semi-linear in the second argument and which satisfies fV (ēi, ēj) = δij for all i, j ∈ {1, . . . , 2n}. With the pair (V, fV ), there is associated a Hermitian dual polar space DH(V,B) ∼= DH(2n − 1, 4), and every point of Gn(V,B) is also a point of DH(V,B). By [10] or [11, Section 6.3], the set X of points of Gn(V,B) is a subspace of DH(V,B) and the following two properties hold: the electronic journal of combinatorics 16 (2009), #R137 2 (1) X̃ = Gn(V,B); (2) If x and y are two points of X, then the distance between x and y in X̃ equals the distance between x and y in DH(V,B). Properties (1) and (2) imply that the near polygon Gn admits a full and isometric embedding into the dual polar space DH(2n− 1, 4). It can be shown that there exists up to isomorphism a unique such isometric embedding, see De Bruyn [16]. Suppose S = (P,L, I) is a dense near polygon. A function f : P → N is called a valuation of S if it satisfies the following properties: