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David Roberts
David Roberts
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$O(2n)$ Orbits in $\Lambda^{2k}(\mathbb{R}^{2n})$ Classification of $2k$-vectors modulo orthogonal transformations

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David Roberts
David Roberts
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Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of equivalence classesorbit spaces of even-rank anti-symmetric tensors, where $$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}\mapsto Re_{i_1}\wedge \cdots \wedge Re_{i_{2k}}},~~~~~~~~~R\in O(2n),$$ The base case is well-known, and corresponds to the canonical form for a real, antisymmetric matrix: $$A_1=\frac{\Lambda^2(\mathbb{R}^{2n})}{e_i\wedge e_j\mapsto Re_i\wedge Re_j}=\left\{\bigg[\sum_{j}\lambda_{j} e_{2j-1}\wedge e_{2j}\bigg],~~\lambda_j\in\mathbb{R}\right\}\simeq \mathbb{R}^n$$ Are there any results for $A_2$?

Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of equivalence classes of even-rank anti-symmetric tensors, where $$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}\mapsto Re_{i_1}\wedge \cdots \wedge Re_{i_{2k}}},~~~~~~~~~R\in O(2n),$$ The base case is well-known, and corresponds to the canonical form for a real, antisymmetric matrix: $$A_1=\frac{\Lambda^2(\mathbb{R}^{2n})}{e_i\wedge e_j\mapsto Re_i\wedge Re_j}=\left\{\bigg[\sum_{j}\lambda_{j} e_{2j-1}\wedge e_{2j}\bigg],~~\lambda_j\in\mathbb{R}\right\}\simeq \mathbb{R}^n$$ Are there any results for $A_2$?

Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where $$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}\mapsto Re_{i_1}\wedge \cdots \wedge Re_{i_{2k}}},~~~~~~~~~R\in O(2n),$$ The base case is well-known, and corresponds to the canonical form for a real, antisymmetric matrix: $$A_1=\frac{\Lambda^2(\mathbb{R}^{2n})}{e_i\wedge e_j\mapsto Re_i\wedge Re_j}=\left\{\bigg[\sum_{j}\lambda_{j} e_{2j-1}\wedge e_{2j}\bigg],~~\lambda_j\in\mathbb{R}\right\}\simeq \mathbb{R}^n$$ Are there any results for $A_2$?

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David Roberts
David Roberts
  • 605
  • 3
  • 11

$O(2n)$ Orbits in $\Lambda^{2k}(\mathbb{R}^{2n})$

Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of equivalence classes of even-rank anti-symmetric tensors, where $$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}\mapsto Re_{i_1}\wedge \cdots \wedge Re_{i_{2k}}},~~~~~~~~~R\in O(2n),$$ The base case is well-known, and corresponds to the canonical form for a real, antisymmetric matrix: $$A_1=\frac{\Lambda^2(\mathbb{R}^{2n})}{e_i\wedge e_j\mapsto Re_i\wedge Re_j}=\left\{\bigg[\sum_{j}\lambda_{j} e_{2j-1}\wedge e_{2j}\bigg],~~\lambda_j\in\mathbb{R}\right\}\simeq \mathbb{R}^n$$ Are there any results for $A_2$?