1,721,018 research outputs found
On Hermitian spreads
No full textLet ⊥ be the polarity of PG(5, q) defined by the elliptic quadric Q- (5, q). A locally Hermitian spread S of Q- (5, q), with respect to a line L, is associated in a canonical way with a spread S B of the 3-dimensional projective space L⊥ = Λ, and conversely. In this paper we give a geometric characterization of the regular spreads of Λ which induce Hermitian spreads of Q- (5, q).Let perpendicular to be the polarity of PG(5, q) defined by the elliptic quadric Q(-) (5, q). A locally Hermitian spread S of Q(-) (5, q), with respect to a line L, is associated in a canonical way with a spread S-Lambda of the 3-dimensional projective space L-perpendicular to = Lambda, and conversely. In this paper we give a geometric charaterization of the regular spreads of Lambda which induce Hermitian spreads of Q(-) (5, q)
On the Grassmann module of symplectic dual polar spaces of rank 4 in characteristic 3
No full textLet V be the Weyl module of dimension (2n n) - (2n n-2) for the symplectic group Sp(2n, F) whose highest weight is the nth fundamental dominant weight. The module V affords the grassmann embedding of the symplectic dual polar space DW (2n - 1, F), therefore V is also called the grassmann module for the symplectic group.
We consider the smallest case for char(F) odd for which V is reducible, namely n = 4 and char(F) = 3. In this case the unique factor R of V has vector dimension I. Here we provide a geometric description for Rand study some relations between Rand other objects associated with the grassmann embedding. (C) 2009 Elsevier B.V. All rights reserved
On certain submodules of Weyl modules for SO(2n+1,F) with char(F) = 2
No full textFor let be the Weyl module for the special orthogonal group G = \mathrm{SO}(2n+1,\F) with respect to the -th fundamental dominant weight of the root system of type and put . It is well known that all of these modules are irreducible when \mathrm{char}(\F) \neq 2 while when \mathrm{char}(\F) = 2 they admit many proper submodules. In this paper, assuming that \mathrm{char}(\F) = 2, we prove that admits a chain of submodules where for and is the trivial 1-dimensional module. We also show that for the quotient is isomorphic to the so called -th Grassmann module for . Resting on this fact we can give a geometric description of as a submodule of the -th Grassmann module. When \F is perfect G\cong \mathrm{Sp}(2n,\F) and is isomorphic to the Weyl module for \mathrm{Sp}(2n,\F) relative to the -th fundamental dominant weight of the root system of type . All irreducible sections of the latter modules are known. Thus, when \F is perfect, all irreducible sections of are known as well
Some results on caps and codes related to orthogonal Grassmannians — a preview
No full textIn this note we offer a short summary of some recent results, to be contained in
a forthcoming paper [4], on projective caps and linear error correcting codes arising from the Grassmann embedding εgr
k of an orthogonal Grassmannian ∆k . More
precisely, we consider the codes arising from the projective system determined by
εgr
k (∆k ) and determine some of their parameters. We also investigate special sets
of points of ∆k which are met by any line of ∆k in at most 2 points proving that
their image under the Grassmann embedding is a projective cap
Veronesean embeddings of dual polar spaces of orthogonal type
No full textGiven a point-line geometry and a pappian projective space , a veronesean embedding of in is an injective map from the point-set of to the set of points of mapping the lines of onto non-singular conics of and such that spans . In this paper we study veronesean embeddings of the dual polar space associated to a non-singular quadratic form of Witt index in . Three such embeddings are considered, namely the Grassmann embedding which maps a maximal singular subspace of (namely a point of ) to the point of , the composition of the spin (projective) embedding of in with the quadric veronesean map , and a third embedding defined algebraically in the Weyl module , where is the fundamental dominant weight associated to the -th simple root of the root system of type . We shall prove that and are isomorphic. If \mathrm{char}(\F)\neq 2 then is irreducible and is isomorphic to while if \mathrm{char}(\F)= 2 then is a proper quotient of . In this paper we shall study some of these submodules. Finally we turn to universality, focusing on the case of . We prove that if \F is a finite field of odd order then is relatively universal. On the contrary, if \mathrm{char}(\F)= 2 then is not universal. We also prove that if \F is a perfect field of characteristic 2 then is not universal, for any
A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3
No full textWe give a geometrical description of the spin-embedding esp of the symplectic dual polar space Δ ≅ DW (5, 2r) by showing how the natural embedding of W (5, 2r) into PG (5, 2r) is involved in the Grassmann-embedding egr of Δ. We prove that the map sending every quad of Δ to its nucleus realizes the natural embedding of W (5, 2r). Taking the quotient of egr over the space spanned by the nuclei of the quadrics corresponding to the quads of Δ gives an embedding isomorphic to esp. © 2007 Elsevier Inc. All rights reserved
The structure of full polarized embeddings of symplectic and hermitian dual polar spaces
No full textLet Delta be a thick dual polar space of rank n >= 2 and let e be a full polarized embedding of Delta into a projective space Sigma. For every point x of Delta and every i is an element of {0,..., n}, let T-i(x) denote the subspace of Sigma generated by all points e(y) with d(x, y) = i + 1. We also show that there exists a well-defined map e(i)(x) from the set of (i - 1)-dimensional subspaces of the residue Res(Delta)(x) of Delta at the point x (which is a projective space of dimension n - 1) to the set of points of the quotient space T-i(x)/Ti-1(x). In this paper we study the structure of the maps e(i)(x) and the subspaces T-i(x) for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature
On certain forms and quadrics related to symplectic dual polar spaces in characteristic 2
No full textLet be a -dimensional vector space over a field \F and a non-degenerate alternating form defined on . Let be the building of type formed by the totally -isotropic subspaces of and, for , let \G_k and be the -grassmannians of \PG(V) and , embedded in and in a subspace respectively, where . This paper is a continuation of \cite{CPeven2}. In \cite{CPeven2}, focusing on the case of , we considered two forms and related to the notion of \lq being at non maximal distance\rq\, in \G_n and and, under the hypothesis that \mathrm{char}(\F) \neq 2, we studied the subspaces of where and coincide or are opposite. In this paper we assume that \mathrm{char}(\F)=2. We determine which of the quadrics associated to or are preserved by the group G= \mathrm{Sp}(2n,\F) in its action on and we study the subspace of formed by vectors such that for every . Finally, we show how properties of can be exploited to investigate the poset of -invariant subspaces of for and
Semifield planes of order q4 with kernel Fq2 and center Fq
No full textA classification of semifield planes of order with kernel
and center is given. For odd prime, this
proves the conjecture stated in by M.~Cordero and
R.~Figueroa. Also, we extend the classification of
semifield planes lifted from Desarguesian planes of order odd, obtained by Cordero and Figueroa in , to the
even characteristic case
Liturgica 1 Cardinali I. A. Schuster in memoriam (Scripta et Documenta 7), 1956
No full textChirat Henri. Liturgica 1 Cardinali I. A. Schuster in memoriam (Scripta et Documenta 7), 1956. In: Revue des Sciences Religieuses, tome 32, fascicule 3, 1958. pp. 298-302
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