The pseudo-hyperplanes and homogeneous pseudo-embeddings of AG(n, 4) and PG(n, 4)

Abstract

We determine all homogeneous pseudo-embeddings of the affine space AG(n, 4) and the projective space PG(n, 4). We give a classification of all pseudo-hyperplanes of AG(n, 4). We also prove that the two homogeneous pseudo-embeddings of the generalized quadrangle Q(4, 3) are induced by the two homogeneous pseudo-embeddings of AG(4, 4) into which Q(4, 3) is fully embeddable.Keywords: homogeneous pseudo-embedding, pseudo-hyperplane MSC2000: 51E20, 05B25 Basic definitions and main resultsThe aim of this section is to state the main results of this paper and to define the basic notions which are necessary to understand these results. Throughout this section, S = (P, L, I) is a point-line geometry with the property that the number of points on each line is finite and at least three.Suppose V is a vector space over the field F 2 of order 2. A pseudo-embedding of S into the projective space Σ = PG(V ) is a mapping e from P to the point set of Σ satisfying: (1) < e(P) > Σ = Σ; (2) if L is a line of S with points x 1 , x 2 , . . . , x k , then the points e(x 1 ), e(x 2 ), . . . , e(x k−1 ) of Σ are linearly independent and e(x k ) =<v 1 +v 2 + · · · +v k−1 > wherev i , i ∈ {1, 2, . . . , k − 1}, is the unique vector of V for which e(x i ) =<v i > Σ . Two pseudo-embeddings e 1 : S → Σ 1 and e 2 : S → Σ 2 of S are called isomorphic (e 1 ∼ = e 2 ) if there exists an isomorphism φ : Σ 1 → Σ 2 such that e 2 = φ • e 1 . The notion pseudoembedding was introduced in De Bruyn [1].Suppose e : S → PG(V ) is a pseudo-embedding of S and G is a group of automorphisms of S. We say that e is G-homogeneous if for every θ ∈ G, there exists a (necessarily unique) projectivity η θ of PG(V ) such that e(x θ ) = e(x) η θ for every point x of S. If G is the full automorphism group of S, then e is also called a homogeneous pseudo-embedding.Suppose e : S → Σ is a pseudo-embedding of S and α is a subspace of Σ satisfying the following two properties: 1

Veronese Representation of Projective Hjelmslev Planes Over Some Quadratic Alternative Algebras

“…Indeed, as a side result, our axioms provide a new characterisation of the above mentioned nondegenerate varieties, except that over the field of order 2 more examples pop up (still classifiable though), and interestingly, one example is strongly related to the large Witt design S(24, 5,8) and the sporadic Mathieu group M 24 . These extra examples are so-called pseudo-embeddings and our results complement in a surprising way the results of De Bruyn [1,2].…”

“…Pseudo embedding of point-line geometries have been introduced and studied by De Bruyn [1,2]. In Proposition 4.1 of [2], he obtained that the universal pseudoembeddings of PG(2, 4) lives in PG(10, 2) and an explicit (coordinate) construction has been given by him in Theorem 1.1 of [1]. Nevertheless we include our construction, which is in terms of a basis of PG(10, 2) because we will rely on it in the lemmas thereafter to prove results in our more specific setting (in which (MM2) also holds).…”

“…Since 6, 8 do not occur in any such triple, the remaining blocks are [1,6] and [1,8], and they need to be distinct (there is no * -triple neither containing 1 nor 6 nor 8). Hence the block [1,6] has to contain x-triples not containing 1 and 6, but there are exactly three such. Consequently, there is only one possibility: [1,6] A straightforward verification shows that each such block sums up to zero.…”

“…Hence the block [1,6] has to contain x-triples not containing 1 and 6, but there are exactly three such. Consequently, there is only one possibility: [1,6] A straightforward verification shows that each such block sums up to zero.…”

The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t)

“…In [9, Theorem 1.7], we will show that the two homogeneous pseudo-embeddings of Q(4, 3) are induced by the two homogeneous pseudo-embeddings of AG(4, 4) into which Q(4, 3) is fully embeddable. In [9], also explicit constructions will be given for the two homogeneous pseudo-embeddings of AG(n, 4), n ≥ 2. Using this, it is thus possible to give explicit constructions for the two homogeneous pseudo-embeddings of Q(4, 3), as well as two of the five homogeneous pseudo-embeddings of GQ (3,5).…”