On certain forms and quadrics related to symplectic dual polar spaces in characteristic 2 |
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Authors: | Ilaria Cardinali Antonio Pasini |
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Institution: | 1. Department of Information Engineering, University of Siena, Via Roma 56, 53100, Siena, Italy 2. Department of Mathematics, University of Siena, Pian dei Mantellini 44, 53100, Siena, Italy
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Abstract: | Let V be a 2n-dimensional vector space over a field ${\mathbb {F}}$ and ξ a non-degenerate alternating form defined on V. Let Δ be the building of type C n formed by the totally ξ-isotropic subspaces of V and, for 1 ≤ k ≤ n, let ${\mathcal {G}_k}$ and Δ k be the k-grassmannians of PG(V) and Δ, embedded in ${W_k=\wedge^kV}$ and in a subspace ${V_k\subseteq W_k}$ respectively, where ${{\rm dim}(V_k)={2n \choose k} - {2n \choose {k-2}}}$ . This paper is a continuation of Cardinali and Pasini (Des. Codes. Cryptogr., to appear). In Cardinali and Pasini (to appear), focusing on the case of k = n, we considered two forms α and β related to the notion of ‘being at non maximal distance’ in ${\mathcal {G}_n}$ and Δ n and, under the hypothesis that ${{\rm char}(\mathbb {F}) \neq 2}$ , we studied the subspaces of W n where α and β coincide or are opposite. In this paper we assume that ${{\rm char}(\mathbb {F}) = 2}$ . We determine which of the quadrics associated to α or β are preserved by the group ${G= {\rm Sp}(2n, \mathbb {F})}$ in its action on W n and we study the subspace ${\mathcal {D}}$ of W n formed by vectors v such that α(v, x) = β(v, x) for every ${x \in W_n}$ . Finally, we show how properties of ${\mathcal {D}}$ can be exploited to investigate the poset of G-invariant subspaces of V k for k = n ? 2i and ${1\leq i \leq \lfloor n/2\rfloor}$ . |
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