A characterization of a class of hyperplanes of

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• 作者：Bart De Bruyn ^{bdb@cage.ugent.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Symplectic dual polar space ; Hyperplane (of subspace-type) ; Spin-embedding
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：6 January 2017
• 年：2017
• 卷：340
• 期：1
• 页码：3176-3182
• 全文大小：421 K

文摘

A hyperplane of the symplectic dual polar space 0f9da144bfc69630b38688876bc4c52" title="Click to view the MathML source">DW(2n−1,F)$DW(2n-1,\mathbb{F})$, n≥2$n\ge 2$, is said to be of subspace-type if it consists of all maximal singular subspaces of W(2n−1,F)$W(2n-1,\mathbb{F})$ meeting a given (n−1)$(n-1)$-dimensional subspace of 208a490b4329a368ef0c4d7ab16" title="Click to view the MathML source">PG(2n−1,F)$\text{PG}(2n-1,\mathbb{F})$. We show that a hyperplane of 0f9da144bfc69630b38688876bc4c52" title="Click to view the MathML source">DW(2n−1,F)$DW(2n-1,\mathbb{F})$ is of subspace-type if and only if every hex 0f1d2f782679f261892fc1" title="Click to view the MathML source">F$F$ of 0f9da144bfc69630b38688876bc4c52" title="Click to view the MathML source">DW(2n−1,F)$DW(2n-1,\mathbb{F})$ intersects it in either 0f1d2f782679f261892fc1" title="Click to view the MathML source">F$F$, a singular hyperplane of 0f1d2f782679f261892fc1" title="Click to view the MathML source">F$F$ or the extension of a full subgrid of a quad. In the case F$\mathbb{F}$ is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane H$H$ of 0f9da144bfc69630b38688876bc4c52" title="Click to view the MathML source">DW(2n−1,F)$DW(2n-1,\mathbb{F})$ is of subspace-type or arises from the spin-embedding of f45af289c61072" title="Click to view the MathML source">DW(2n−1,F)≅DQ(2n,F)$DW(2n-1,\mathbb{F})\cong DQ(2n,\mathbb{F})$ if and only if every hex 0f1d2f782679f261892fc1" title="Click to view the MathML source">F$F$ intersects it in either 0f1d2f782679f261892fc1" title="Click to view the MathML source">F$F$, a singular hyperplane of 0f1d2f782679f261892fc1" title="Click to view the MathML source">F$F$, a hexagonal hyperplane of 0f1d2f782679f261892fc1" title="Click to view the MathML source">F$F$ or the extension of a full subgrid of a quad.