Let k be a field with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\text {char}(k)\ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>char</mtext> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove that all maximal flags of composition algebras over k, appear as the k-rational <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Sp_{6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>p</mi> <mn>6</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-orbits in a Zariski-dense <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Sp_{6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>p</mi> <mn>6</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-invariant subset <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$V^{ss}\subset V=\wedge ^{3}V_{6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>V</mi> <mrow> <mi mathvariant="italic">ss</mi> </mrow> </msup> <mo>⊂</mo> <mi>V</mi> <mo>=</mo> <msup> <mo>∧</mo> <mn>3</mn> </msup> <msub> <mi>V</mi> <mn>6</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$V_{6}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mn>6</mn> </msub> </math></EquationSource> </InlineEquation> is the standard 6-dimensional irreducible representation of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$Sp_{6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>p</mi> <mn>6</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(Sp_{6}\times GL_{1}^{2},V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <msub> <mi>p</mi> <mn>6</mn> </msub> <mo>×</mo> <mi>G</mi> <msubsup> <mi>L</mi> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(GSp_{6}\times GL_{1}^{2},V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mi>S</mi> <msub> <mi>p</mi> <mn>6</mn> </msub> <mo>×</mo> <mi>G</mi> <msubsup> <mi>L</mi> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We also get all reduced Freudenthal algebras of dimensions 6 and 9, represented by the same orbit spaces.

Rational Orbits in Some Prehomogeneous Vector Spaces Associated to $Sp_{6}$ Revisited

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发表日期：

2026年6月23日

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Sayan Pal

摘要

Let k be a field with $\text {char}(k)\ne 2$ $char (k) \neq 2$ . We prove that all maximal flags of composition algebras over k, appear as the k-rational $Sp_{6}$ $S p_{6}$ -orbits in a Zariski-dense $Sp_{6}$ $S p_{6}$ -invariant subset $V^{ss}\subset V=\wedge ^{3}V_{6}$ $V^{ss} \subset V = \land^{3} V_{6}$ , where $V_{6}$ $V_{6}$ is the standard 6-dimensional irreducible representation of $Sp_{6}$ $S p_{6}$ . This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces $(Sp_{6}\times GL_{1}^{2},V)$ $(S p_{6} \times G L_{1}^{2}, V)$ and $(GSp_{6}\times GL_{1}^{2},V)$ $(G S p_{6} \times G L_{1}^{2}, V)$ . We also get all reduced Freudenthal algebras of dimensions 6 and 9, represented by the same orbit spaces.

Rational Orbits in Some Prehomogeneous Vector Spaces Associated to \(Sp_{6}\) Revisited

摘要