The main problem considered in this paper is “how does a dual polar space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> of rank 3 embed in a metasymplectic space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation>?” The expected and generic answer is that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is isomorphic to a subgeometry of a point residual <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textsf{Res}_\Delta (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="sans-serif">Res</mi> <mi mathvariant="normal">Δ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and that it arises as a subgeometry of a trace geometry, that is, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Gamma \subseteq p^\perp \cap q^{\bowtie }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>⊆</mo> <msup> <mi>p</mi> <mo>⊥</mo> </msup> <mo>∩</mo> <msup> <mi>q</mi> <mo>⋈</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, for two opposite points p and q, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$q^{\bowtie }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mo>⋈</mo> </msup> </math></EquationSource> </InlineEquation> is the set of points special to q. However, this is not always the case, and we describe some counterexamples, even classify them for certain classes of metasymplectic spaces <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation>. These results complement the analogous results for the exceptional geometries of diameter at most 3 arising from groups of types <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathsf {E_6,E_7,E_8}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="sans-serif">E</mi> <mn mathvariant="sans-serif">6</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="sans-serif">E</mi> <mn mathvariant="sans-serif">7</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="sans-serif">E</mi> <mn mathvariant="sans-serif">8</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> recently treated by Cooperstein and the second author.

Dual polar spaces embedded in metasymplectic spaces

期刊论文
发表日期：

2026年3月27日

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Linde Lambrecht,
Hendrik Van Maldeghem

摘要

The main problem considered in this paper is “how does a dual polar space $\Gamma $ $Γ$ of rank 3 embed in a metasymplectic space $\Delta $ $Δ$ ?” The expected and generic answer is that $\Gamma $ $Γ$ is isomorphic to a subgeometry of a point residual $\textsf{Res}_\Delta (p)$ ${Res}_{Δ} (p)$ and that it arises as a subgeometry of a trace geometry, that is, $\Gamma \subseteq p^\perp \cap q^{\bowtie }$ $Γ \subseteq p^{⊥} \cap q^{⋈}$ , for two opposite points p and q, where $q^{\bowtie }$ $q^{⋈}$ is the set of points special to q. However, this is not always the case, and we describe some counterexamples, even classify them for certain classes of metasymplectic spaces $\Delta $ $Δ$ . These results complement the analogous results for the exceptional geometries of diameter at most 3 arising from groups of types $\mathsf {E_6,E_7,E_8}$ $E_{6}, E_{7}, E_{8}$ recently treated by Cooperstein and the second author.