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hyperplane arrangement
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(Definition)
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Let $V$ be a vector space over a field $\mathbb{K}$ A hyperplane arrangment in $V$ is a family $\mathcal{A}=\{\mathcal{H}_i\}_{i\in I}$ of affine hyperplanes in $V$ If all of the hyperplanes pass through $0$ $\mathcal{A}$ is called central; otherwise, it is affine. More generally, a
subspace arrangement is a family of affine subspaces of $V$ The same distinction between central and affine subspace arrangement holds.
Example 1 Let $V=\mathbb{K}^n$ Then the family $$ \mathbb{K}P^n=\{S\subset V\mid\dim_{\mathbb{K}}(S)=1\} $$ of $1$ dimensional subspaces of $V$ is a central subspace arrangement, the projective space of dimension $n$ over $\mathbb{K}$
Instead of considering all lines through a vector space, we could consider all $k$ dimensional subspaces of the space.
Example 2 Again let $V=\mathbb{K}^n$ and suppose $0\le k\le n$ Then the family $$ \Gr(V,k)=\{S\subset V\mid\dim_{\mathbb{K}}(S)=k\} $$ of $k$ dimensional subspaces of $V$ is a central subspace arrangement, the Grassmannian. Observe that $\mathbb{K}P^n=\Gr(\mathbb{K}^n,1)$
If $V$ is a topological vector space and $\mathcal{A}$ is a hyperplane arrangement, then it makes sense to ask for the fundamental group of the complement $V\setminus\bigcup_{\mathcal{H}\in\mathcal{A}}\mathcal{H}$
Example 3 If $\mathcal{A}$ is a finite hyperplane arrangement over $V=\mathcal{R}^n$ then the arrangement partitions $V$ into a finite number of contractible cells. By selecting a point in each cell and taking the convex
hull of the result, we obtain a polytope combinatorially equivalent to the zonotope dual to the arrangement. Since the question of the fundamental group here is not interesting, we could also use the embedding $\mathbb{R}^n\hookrightarrow\mathbb{C}^n$ to complexify $\mathcal{A}$ In this case the complement $\mathbb{C}^n\setminus\bigcup_{\mathcal{H}\in\mathcal{A}}\mathcal{H}$ usually has nontrivial fundamental group.
- 1
- Klain, D. A., and G.-C. Rota, , Introduction to geometric probability, Cambridge University Press, 1997.
- 2
- Orlik, P., and H. Terao, Arrangements of hyperplanes, Springer-Verlag, 1992.
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"hyperplane arrangement" is owned by mps.
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See Also: zonotope
| Other names: |
subspace arrangement, central arrangement |
| Also defines: |
Grassmannian |
| Keywords: |
arrangement, partition, fundamental group |
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Cross-references: embedding, zonotope, polytope, convex hull, point, cells, contractible, number, finite, complement, fundamental group, topological vector space, lines, projective space, subspaces, affine subspaces, pass through, affine hyperplanes, hyperplane, field, vector space
There are 4 references to this entry.
This is version 3 of hyperplane arrangement, born on 2006-03-23, modified 2006-03-23.
Object id is 7760, canonical name is HyperplaneArrangement.
Accessed 4014 times total.
Classification:
| AMS MSC: | 52C35 (Convex and discrete geometry :: Discrete geometry :: Arrangements of points, flats, hyperplanes) | | | 52B99 (Convex and discrete geometry :: Polytopes and polyhedra :: Miscellaneous) |
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Pending Errata and Addenda
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