1,194 research outputs found

    New constructions of two slim dense near hexagons

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    We provide a geometrical construction of the slim dense near hexagon with parameters (s,t,t2)=(2,5,{1,2})(s,t,t_{2})=(2,5,\{1,2\}). Using this construction, we construct the rank 3 symplectic dual polar space DSp(6,2)DSp(6,2) which is the slim dense near hexagon with parameters (s,t,t2)=(2,6,2)(s,t,t_{2})=(2,6,2). Both the near hexagons are constructed from two copies of a generalized quadrangle with parameters (2,2)

    Dual embeddings of dense near polygons

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    Let e: S -> Sigma be a full polarized projective embedding of a dense near polygon S, i.e., for every point p of S, the set H(p) of points at non-maximal distance from p is mapped by e into a hyperplane Pi(p) of Sigma. We show that if every line of S is incident with precisely three points or if S satisfies a certain property (P(de)) then the map p bar right arrow Pi p defines a full polarized embedding e* (the so-called dual embedding of e) of S into a subspace of the dual Sigma* of Sigma. This generalizes a result of [6] where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P(de)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding

    Polarized non-abelian representations of slim near-polar spaces

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    In (Bull Belg Math Soc Simon Stevin 4:299-316, 1997), Shult introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in Sahoo and Sastry (J Algebraic Comb 29:195-213, 2009) for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding

    On the order of a non-abelian representation group of a slim dense near hexagon

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    We show that, if the representation group RR of a slim dense near hexagon SS is non-abelian, then RR is of exponent 4 and ∣R∣=2β|R|=2^{\beta}, 1+NPdim(S)≤β≤1+dimV(S)1+NPdim(S)\leq \beta\leq 1+dimV(S), where NPdim(S)NPdim(S) is the near polygon embedding dimension of SS and dimV(S)dimV(S) is the dimension of the universal representation module V(S)V(S) of SS. Further, if β=1+NPdim(S)\beta =1+NPdim(S), then RR is an extraspecial 2-group (Theorem 1.6)

    Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties

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    Let S(N)≡PG(1, 2)×PG(1, 2)×⋯×PG(1, 2)S_{(N)} \equiv PG(1,\,2) \times PG(1,\,2) \times \cdots \times PG(1,\,2) be a Segre variety that is NN-fold direct product of projective lines of size three. Given two geometric hyperplanes H′H' and H′′H'' of S(N)S_{(N)}, let us call the triple {H′,H′′,H′ΔH′′‾}\{H', H'', \overline{H' \Delta H''}\} the Veldkamp line of S(N)S_{(N)}. We shall demonstrate, for the sequence 2≤N≤42 \leq N \leq 4, that the properties of geometric hyperplanes of S(N)S_{(N)} are fully encoded in the properties of Veldkamp {\it lines} of S(N−1)S_{(N-1)}. Using this property, a complete classification of all types of geometric hyperplanes of S(4)S_{(4)} is provided. Employing the fact that, for 2≤N≤42 \leq N \leq 4, the (ordinary part of) Veldkamp space of S(N)S_{(N)} is PG(2N−1,2)PG(2^N-1,2), we shall further describe which types of geometric hyperplanes of S(N)S_{(N)} lie on a certain hyperbolic quadric Q0+(2N−1,2)⊂PG(2N−1,2)\mathcal{Q}_0^+(2^N-1,2) \subset PG(2^N-1,2) that contains the S(N)S_{(N)} and is invariant under its stabilizer group; in the N=4N=4 case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type LG(4,8)LG(4,8), to the set of 2295 maximal subspaces of the symplectic polar space W(7,2)\mathcal{W}(7,2).Comment: 16 pages, 8 figures and 7 table

    Dense near octagons with four points on each line, III

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    This is the third paper dealing with the classification of the dense near octagons of order (3, t). Using the partial classification of the valuations of the possible hexes obtained in [12], we are able to show that almost all such near octagons admit a big hex. Combining this with the results in [11], where we classified the dense near octagons of order (3, t) with a big hex, we get an incomplete classification for the dense near octagons of order (3, t): There are 28 known examples and a few open cases. For each open case, we have a rather detailed description of the structure of the near octagons involved

    A note on near hexagons with lines of size 3

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    We classify all finite near hexagons which satisfy the following properties for a certain t(2) is an element of {1, 2, 4}: (i) every line is incident with precisely three points; (ii) for every point x, there exists a point y at distance 3 from x; (iii) every two points at distance 2 from each other have either 1 or t(2) + 1 common neighbours; (iv) every quad is big. As a corollary, we obtain a classification of all finite near hexagons satisfying (i), (ii) and (iii) with t(2) equal to 4

    On semi-finite hexagons of order (2,t)(2, t) containing a subhexagon

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    The research in this paper was motivated by one of the most important open problems in the theory of generalized polygons, namely the existence problem for semi-finite thick generalized polygons. We show here that no semi-finite generalized hexagon of order (2,t)(2,t) can have a subhexagon HH of order 22. Such a subhexagon is necessarily isomorphic to the split Cayley generalized hexagon H(2)H(2) or its point-line dual HD(2)H^D(2). In fact, the employed techniques allow us to prove a stronger result. We show that every near hexagon S\mathcal{S} of order (2,t)(2,t) which contains a generalized hexagon HH of order 22 as an isometrically embedded subgeometry must be finite. Moreover, if H≅HD(2)H \cong H^D(2) then S\mathcal{S} must also be a generalized hexagon, and consequently isomorphic to either HD(2)H^D(2) or the dual twisted triality hexagon T(2,8)T(2,8).Comment: 21 pages; new corrected proofs of Lemmas 4.6 and 4.7; earlier proofs worked for generalized hexagons but not near hexagon
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