Calculating the Dimension of the Universal Embedding of the Symplectic Dual Polar Space using Languages

The main result of this paper is the construction of a bijection of the set of words in so-called standard order of length $n$ formed by four different letters and the set $\mathcal{N}^n$ of all subspaces of a fixed $n$-dimensional maximal isotropic subspace of the $2n$-dimensional symplectic space $V$ over $\mathbb{F}_2$ which are not maximal in a certain sense. Since the number of different words in standard order is known, this gives an alternative proof for the formula of the dimension of the universal embedding of a symplectic dual polar space $\mathcal{G}_n$. Along the way, we give formulas for the number of all $n$- and $(n-1)$-dimensional totally isotropic subspaces of $V$.

Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n − 1, 2)

We classify all homogeneous pseudo-embeddings of the point-line geometry defined by the points and k-dimensional subspaces of PG(n, 2), and use this to study the local structure of homogeneous full projective embeddings of the dual polar space DW(2n − 1, 2). Our investigation allows us to distinguish n possible types for such homogeneous embeddings. For each of these n types, we construct a homogeneous full projective embedding of DW(2n − 1, 2).

The hyperplanes of DW(5,F) arising from the Grassmann embedding

The hyperplanes of the symplectic dual polar space DW(5; F) that arise from the Grassmann embedding have been classied in [B.N. Cooperstein and B. De Bruyn. Points and hyperplanes of the universal embedding space of the dual polar space DW(5; q), q odd. Michigan Math. J., 58:195{212, 2009.] in case F is a finite field of odd characteristic, and in [B. De Bruyn. Hyperplanes of DW(5;K) with K a perfect eld of characteristic 2. J. Algebraic Combin., 30:567{584, 2009.] in case F is a perfect eld of characteristic 2. In the present paper, these classifications are extended to arbitrary fields. In the case of characteristic 2 however, it was not possible to provide a complete classification. The main tool in the proof is the classification of the quasi-Sp(V; f)-equivalence classes of trivectors of a 6-dimensional symplectic vector space (V; f) obtained in [B. De Bruyn and M. Kwiatkowski. A 14-dimensional module for the symplectic group: orbits on vectors. Comm. Algebra,43:4553{4569, 2015.

Non-Classical Hyperplanes of $DW(5,q)$

The hyperplanes of the symplectic dual polar space $DW(5,q)$ arising from embedding, the so-called classical hyperplanes of $DW(5,q)$, have been determined earlier in the literature. In the present paper, we classify non-classical hyperplanes of $DW(5,q)$. If $q$ is even, then we prove that every such hyperplane is the extension of a non-classical ovoid of a quad of $DW(5,q)$. If $q$ is odd, then we prove that every non-classical ovoid of $DW(5,q)$ is either a semi-singular hyperplane or the extension of a non-classical ovoid of a quad of $DW(5,q)$. If $DW(5,q)$, $q$ odd, has a semi-singular hyperplane, then $q$ is not a prime number.

The Valuations of the Near Polygon ${\Bbb G}_n$

We show that every valuation of the near $2n$-gon ${\Bbb G}_n$, $n \geq 2$, is induced by a unique classical valuation of the dual polar space $DH(2n-1,4)$ into which ${\Bbb G}_n$ is isometrically embeddable.