194 research outputs found
The uniqueness of the SDPS-set of the symplectic dual polar space ,
SDPS-sets are very nice sets of points in dual polar spaces which themselves carry the structure of dual polar spaces. They were introduced in \cite{DB-V:2} because they gave rise to new valuations and hyperplanes of dual polar spaces. In the present paper, we show that the symplectic dual polar space (4n-1,q), has up to isomorphisms a unique SDPS-set
The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings
AbstractWe characterize the hyperplanes of the dual polar space DW(2n−1,q) which arise from projective embeddings as those hyperplanes H of DW(2n−1,q) which satisfy the following property: if Q is an ovoidal quad, then Q∩H is a classical ovoid of Q. A consequence of this is that all hyperplanes of the dual polar spaces DW(2n−1,4), DW(2n−1,16) and DW(2n−1,p) (p prime) arise from projective embeddings
The hyperplanes of the U (4)(3) near hexagon
Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U (4)(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative
On semi-finite hexagons of order containing a subhexagon
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 can have a subhexagon of order .
Such a subhexagon is necessarily isomorphic to the split Cayley generalized
hexagon or its point-line dual . In fact, the employed
techniques allow us to prove a stronger result. We show that every near hexagon
of order which contains a generalized hexagon of
order as an isometrically embedded subgeometry must be finite. Moreover, if
then must also be a generalized hexagon, and
consequently isomorphic to either or the dual twisted triality hexagon
.Comment: 21 pages; new corrected proofs of Lemmas 4.6 and 4.7; earlier proofs
worked for generalized hexagons but not near hexagon
The hyperplanes of which arise from embedding.
We show that there are 6 isomorphism classes of hyperplanes of the dual polar space which arise from the Grassmann-embedding. If \geq 2\Delta, then there are 6 extra classes of hyperplanes as has been shown by Pralle with the aid of a computer. We will give a computer free proof for this fact. The hyperplanes of (5,q) odd, arising from an embedding will be classified in the forthcoming paper
Polygonal valuations
AbstractWe develop a valuation theory for generalized polygons similar to the existing theory for dense near polygons. This valuation theory has applications for the study and classification of generalized polygons that have full subpolygons as subgeometries
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