Classifying GL\((n,\mathbb{Z})\)-orbits of points and rational subspaces (Q321583)
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scientific article; zbMATH DE number 6638737
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Classifying GL\((n,\mathbb{Z})\)-orbits of points and rational subspaces |
scientific article; zbMATH DE number 6638737 |
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Classifying GL\((n,\mathbb{Z})\)-orbits of points and rational subspaces (English)
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14 October 2016
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GL\((n,\mathbb{Z})\)-orbit
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SL\((n,\mathbb{Z})\)-orbit
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rational polyhedron
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rational simplex
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regular simplex
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desingularization
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Hausdorff measure
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rational measure
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rational subspace
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The authors show that the subgroup of the abelian real group \(\mathbb R\) generated by the coordinates of a point in \(\mathbb R^n\) classifies completely the GL\((n,\mathbb{Z})\)-orbit of this point.NEWLINENEWLINEThey also classify GL\((n,\mathbb{Z})\)-orbits of rational affine subspaces \(F\) of \(\mathbb R^n\): in fact, the authors prove that the dimension of \(F\) and the volume of a special parallelotope associated to \(F\) yields a complete classifier of the GL\((n,\mathbb{Z})\)-orbit of \(F\).
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