5,154 research outputs found
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
On hyperovals of polar spaces
We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon E-3 has up to isomorphism a unique full embedding into the dual polar space DH(5, 4)
The nonexistence of regular near octagons with parameters (s, t, t(2), t(3)) = (2,24,0,8)
Let S be a regular near octagon with s + 1 = 3 points per line, let t + 1 denote the constant number of lines through a given point of S and for every two points x and y at distance i is an element of {2, 3} from each other, let t(i) + 1 denote the constant number of lines through y containing a (necessarily unique) point at distance i - 1 from x. It is known, using algebraic combinatorial techniques, that (t(2), t(3), t) must be equal to either (0, 0, 1), (0, 0, 4), (0, 3, 4), (0, 8, 24), (1, 2, 3), (2, 6, 14) or (4, 20, 84). For all but one of these cases, there is a unique example of a regular near octagon known. In this paper, we deal with the existence question for the remaining case. We prove that no regular near octagons with parameters (s, t, t(2), t(3)) = (2, 24, 0, 8) can exist
Dual embeddings of dense near polygons
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
Characterizations of the Suzuki tower near polygons
In recent work, we constructed a new near octagon from certain
involutions of the finite simple group and showed a correspondence
between the Suzuki tower of finite simple groups, , and the tower of near polygons, . Here we characterize
each of these near polygons (except for the first one) as the unique near
polygon of the given order and diameter containing an isometrically embedded
copy of the previous near polygon of the tower. In particular, our
characterization of the Hall-Janko near octagon is similar to an
earlier characterization due to Cohen and Tits who proved that it is the unique
regular near octagon with parameters , but instead of regularity
we assume existence of an isometrically embedded dual split Cayley hexagon,
. We also give a complete classification of near hexagons of
order and use it to prove the uniqueness result for .Comment: 20 pages; some revisions based on referee reports; added more
references; added remarks 1.4 and 1.5; corrected typos; improved the overall
expositio
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
Geometric contextuality from the Maclachlan-Martin Kleinian groups
There are contextual sets of multiple qubits whose commutation is
parametrized thanks to the coset geometry of a subgroup of
the two-generator free group . One defines
geometric contextuality from the discrepancy between the commutativity of
cosets on and that of quantum observables.It is shown in this
paper that Kleinian subgroups that are
non-compact, arithmetic, and generated by two elliptic isometries and
(the Martin-Maclachlan classification), are appropriate contextuality filters.
Standard contextual geometries such as some thin generalized polygons (starting
with Mermin's grid) belong to this frame. The Bianchi groups
, defined over the imaginary quadratic field
play a special role
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