A characterization of some geometries of Lie type

Cohen, Arjeh M.; Cooperstein, Bruce N.

doi:10.1007/bf00146968

Cited by 60 publications

(69 citation statements)

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“…Moreover, it satisfies the axioms (P3)1: and (P4) (defined in [11]) fork= 3. For convenience, we quote them here.…”

Section: Two App LI Cations; Proof Of the Main Theoremmentioning

confidence: 97%

“…whose rank 2 residues are generalized digons or projective planes over K; we shall denote this building by D"(K). The truncation 1°· 2 lD"(K) is a parapolar space as defined in Cohen and Cooperstein [ 11), when the objects of type 0 and 2 are viewed as points and lines respectively. This means that it is a connected partial linear space, all of whose lines have at least three points, such that (i) if l is a line and x is a point collinear with two points on /, then x is collinear with all points on l;…”

Section: Two App LI Cations; Proof Of the Main Theoremmentioning

confidence: 99%

“…(P3) 11 For any two points at mutual distance 2, the set of their common neighbors carries the structure of a nondegenerate polar space of rank k. Proof. By Theorem 1 of Cohen and Cooperstein [11], it is immediate that the geometry over {O, 1, 2, 3} described in the statement of the proposition admits a sheaf on r over the set F of all nonempty flags, which is of type Dn + 1 • This implies the proposition.…”

Section: Two App LI Cations; Proof Of the Main Theoremmentioning

confidence: 99%

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Local recognition of tits geometries of classical type

Brouwer

Cohen

1986

Geom Dedicata

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ABSTRACT. A method, based on Tits' work and involving an idea of M. Ronan, is developed in order to recognize certain geometries which are locally buildings of classical type as quotients of buildings. Two applications are treated in detail showing that every finite nearly classical near polygon must be a dual polar spaoe and that in the finite case of Cooperstein's theorem characterizing geometries of Lie type D., the hypotheses can be weakened considerably. INTRODUCTIONAn interesting topic in the theory of incidence systems (i.e. geometries of rank 2) associated with buildings of spherical type is the local recognition problem. This is the question whether such an incidence system is fully determined by the structure induced on the set of points collinear with a given point. Johnson and Shult [14] is based) is both local and global in nature since it concerns an axiom on incidence systems that forces all distances in the collinearity graph to be at most 2. Two more beautiful characterizations of incidence systems associated with buildings are the theorems by Cameron [9] and Cooperstein [12] on dual polar spaces (type C,.) and halved dual polar spaces (type D,.), respectively. Both, however, use conditions involving points at arbitrarily large mutual distance. The general (and notably Shult's} feeling was that the hypotheses of both theorems might be weakened to conditions involving only points at mutual distance at most m for some suitable natural number m. In [18] Shult describes a way to get rid of at least one of the global assumptions in Cameron's theorem. The main purpose of this article is to establish a version of the theorems of Cameron and of Cooperstein which only uses the part of their

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“…Moreover, it satisfies the axioms (P3)1: and (P4) (defined in [11]) fork= 3. For convenience, we quote them here.…”

Section: Two App LI Cations; Proof Of the Main Theoremmentioning

confidence: 97%

Section: Two App LI Cations; Proof Of the Main Theoremmentioning

confidence: 99%

Section: Two App LI Cations; Proof Of the Main Theoremmentioning

confidence: 99%

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Local recognition of tits geometries of classical type

Brouwer

Cohen

1986

Geom Dedicata

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“…In the majority of cases -in all of them? -we consider the following result is contained in the papers by Bruce Cooperstein and Arjeh Cohen on the characterisation of geometries related to parabolic subgroups of Chevalley groups; see in particular [22,23], and further references in [19]. 2 In particular, in the most interesting case of adjoint representations of type E l , the answer can be found in [23], page 357 for E 6 , page 361 for E 7 , and page 363 for E 8 .…”

Section: Theoremmentioning

confidence: 99%

Numerology of square equations

Vavilov

2009

St. Petersburg Math. J.

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Abstract. In the present work, which is a sequel of the paper "Can one see the signs of structure constants?", we describe how one can see the form and the signs of the senior Weyl orbit of equations on the highest weight orbit directly in the weight diagram of microweight representations and adjoint representations for the simplylaced case. As special cases, the square equations we consider include the vanishing of second order minors, Plücker equations in polyvector and adjoint representations of classical groups, Cartan equations in spin and half-spin representations, BorelFreudenthal equations defining the projective octave plane E 6 /P 1 , and most of the equations defining Freudenthal's variety E 7 /P 7 . In view of forthcoming applications to the construction of decomposition of unipotents in the adjoint case, special emphasis is placed on the senior Weyl orbit of equations for the adjoint representations of groups of types E 6 , E 7 , and E 8 . This orbit consists of 270, 756, or 2160 equations, respectively, and we minutely discuss their form and signs. This generalizes Theorem 3 of the preceding paper "A third look at weight diagrams", where we considered microweight representations of E 6 and E 7 .

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“…This work was thoroughly finite in 1 e.g. the characterization of parapolar spaces by Arjeh Cohen and Bruce Cooperstein [2] 2 e.g. the characterizations of metasymplectic spaces, of buildings of type D 4 and of the exceptional polar spaces of rank three 3 e.g.…”

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confidence: 99%