J. Tits, \Sur la Trialite et Certains Groupes qui s'en Deduisent," Inst. Hautes Etudes Sci. Publ. Math., vol. 2, pp. 14-60, 1959.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....generalized hexagons, generalized octagons, respectively. There is a point line duality for generalized polygons for which in any definition or theorem the words point and line are interchanged and the parameters s and t are interchanged. Generalized polygons were introduced by Tits [1959] in his celebrated paper on triality. There are some equivalent definitions for generalized polygons. Let us mention a rather geometric one (see Van Maldeghem [1998] Let n # 2 be again a natural number. Then a generalized n gon may be defined as a geometry S = P, B, I) with P #= #, B #= ....

J. Tits, Sur la trialite et certains groupes qui s'en deduisent, Inst. Hautes Etudes Sci. Publ. Math. 2 (1959), 13 -- 60.

....Secondly we generalize the flocks associated to the generalized quadrangles K(q) discovered by Kantor [6] We also show its relationship to the split Cayley hexagon H(K)oversomefieldK. 2 Basic definitions and results 2. 1 Generalized quadrangles A generalized quadrangle (GQ) introduced by Tits [13]) is an incidence structure Q = P , B, I) in which P and B are disjoint (nonempty) sets of objects called points and lines respectively, and for which I is a symmetric point line incidence relation satisfying the following properties : GQ1 Two distinct points are incident with at most one ....

J. Tits. Sur la trialite et certains groupes qui s'en deduisent. Publ. Math. IHES, 2, pp. 14--60, 1959.

....these hexagons obvious. Familiarity with the theory of generalised polygons is not assumed. Perhaps this paper will convince the uninitiated reader that even new and seemingly abstract geometrical structures sometimes have nice visual presentations. Generalised polygons were introduced by Tits in [7]. They serve, among other things, as a geometric realisation of certain groups. Generalised triangles are essentially projective planes, generalised quadrangles are also known as polar spaces of rank 2. Classical (thick) generalised n gons, n 2, exist only for n 2 f3; 4; 6; 8g. By [3] also thick ....

....7 gons. In the dual hexagon the orbits not involved in the three ordinary 7 gons do not form easily recognisable figures. To find a cyclic action of order seven and to work out the incidences two different coordinatisations of the G 2 (2) hexagon are used. The first coordinatisation, due to Tits [7], is good for calculations but less practical to detect incidences. The second coordinatisation, introduced by De Smet and Van Maldeghem in [2] is fine to detect incidences. 2. Preliminaries A point line geometry I = P; L; I) consists of a point set P, a line set L and an incidence relation I ....

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J. Tits. Sur la trialit'e et certains groupes qui s'en d'eduisent. Publ. Math. I.H.E.S., 2:13--60, 1959.

....We will view generalized polygons as geometries of rank 2 whose elements are points and lines. The dual is obtained by interchanging these names. A flag is an incident point line pair and hence a chamber in the corresponding spherical rank 2 building. Generalized polygons were introduced by Tits [10] and are the basic rank 2 incidence geometries. Let Gamma be a generalized n gon, n 3. Given a fixed flag F in Gamma, we define Gamma (F ) to be the set of all flags opposite F in Gamma together with all points and lines occurring in these flags. So Gamma (F ) is a rank 2 subgeometry of ....

J. Tits, Sur la trialit'e et certains groupes qui s'en d'eduisent, Inst. Hautes ' Etudes Sci. Publ. Math. 2 (1959), 13 -- 60.

....examples of classifications of non convex embeddings under very mild conditions. Generalized hexagons Here are some remarkable cases. Let us first determine the twisted diagrams of some Moufang hexagons. We consider the hexagons obtained from a triality, as discovered and explained by Tits in [16]. Let first H be the (so called split Cayley) generalized hexagon corresponding to the group G 2 (k) for some field k, where we choose the line set in such a way that it corresponds to a line set of the triality quadric. It is well known, see [16] again, that H lives on a non degenerate quadric ....

....a triality, as discovered and explained by Tits in [16] Let first H be the (so called split Cayley) generalized hexagon corresponding to the group G 2 (k) for some field k, where we choose the line set in such a way that it corresponds to a line set of the triality quadric. It is well known, see [16] again, that H lives on a non degenerate quadric Q(6; k) of maximal Witt index in projective 6 space, and that the points of H collinear to any point p of H are exactly the points of a plane p of Q(6; k) the lines of H are some lines of Q(6; k) Hence we may regard the points of H really as ....

J. Tits, Sur la trialit'e et certains groupes qui s'en d'eduisent, Inst. Hautes ' Etudes Sci. Publ. Math. 2 (1959), 14 -- 60.

....Balaban s graph is not known to be unique. Girth 12 In this case the very naive bound (126 vertices) is attained by a unique graph. The graph is associated with configurations studied by classical geometers such as Edge [20] and it is also implicit in the geometry of triality studied by Tits [46]. The underlying structure is the Lie algebra of type G 2 , which gives rise to the related concept of a generalised hexagon. The first explicitly graph theoretical treatment is due to Benson [3] Girth 13 A Cayley graph with 272 vertices was discovered by Hoare in 1981. It is described in [25] ....

J. Tits. Sur la trialite et certains groupes qui s'en deduisent. Inst. Hautes Etudes Sci. Publ. Math. 2 (1959) 14--60.

.... for instance the split Cayley hexagon has a representation in PG(6; k) such that its points are all points of the quadric with equation X 0 X 4 X 1 X 5 X 2 X 6 = X 2 3 and its lines are all lines whose grassmannian coordinates satisfy six linear equations with coefficients 0, 1 or Gamma1 (see [6]) Hence applying a field automorphism to the coordinates in PG(6; k) induces a collineation of the hexagon. Such a collineation can be chosen to stabilize P and at least three chambers containing p. It follows that the full automorphism group of U P is induced by Gamma. The lemma is proved. Xi ....

J. Tits. Sur la trialit'e et certains groupes qui s'en d'eduisent. Inst. Hautes Etudes Sci. Publ. Math., 2:13 -- 60, 1959.

....Balaban s graph is not known to be unique. Girth 12 In this case the very naive bound (126 vertices) is attained by a unique graph. The graph is associated with configurations studied by classical geometers such as Edge [20] and it is also implicit in the geometry of triality studied by Tits [46]. The underlying structure is the Lie algebra of type G 2 , which gives rise to the related concept of a generalised hexagon. The first explicitly graph theoretical treatment is due to Benson [3] Girth 13 A Cayley graph with 272 vertices was discovered by Hoare in 1981. It is described in [25] ....

J. Tits. Sur la trialit'e et certains groupes qui s'en deduisent. Inst. Hautes Etudes Sci. Publ. Math. 2 (1959) 14--60.

No context found.

J. Tits, \Sur la Trialite et Certains Groupes qui s'en Deduisent," Inst. Hautes Etudes Sci. Publ. Math., vol. 2, pp. 14-60, 1959.

No context found.

J. Tits. Sur la trialite et certains groupes qui s'en deduisent. Publ. Math. I.H.E.S., 2:14--20, 1959.

No context found.

J. Tits. Sur la trialite et certains groupes qui s'en deduisent. Publ. Math. I.H.E.S., 2:14--20, 1959.

No context found.

J. Tits. Sur la trialit'e et certains groupes qui s'en d'eduisent. Publ. Math. I.H.E.S., 2:14--20, 1959.

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