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Coxeter fan
Let \(W\) be a Weyl group.
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The Coxeter arrangement \(\mathcal{A}\) for \(W\) is the collection of all reflecting hyperplanes for \(W\). The complement \(V \backslash(\bigcup \mathcal{A})\) of \(\mathcal{A}\) consists of open cones.
- Their closures are called chambers.
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The chambers are in canonical bijective correspondence with the elements of \(W\). The fundamental chamber \(D:=\bigcap_{s \in S}\left\{v \in V \mathrel{\Big|}\left\langle v, \alpha_{s}\right\rangle \geq 0\right\}\) corresponds to the identity \(e \in W\) and the chamber \(w(D)\) corresponds to \(w \in W\).
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Subsets above and below hyperplanes:
- A subset \(U\) of \(V\) is below a hyperplane \(H \in \mathcal{A}\) if every point in \(U\) is on \(H\) or on the same side of \(H\) as \(D\).
- The subset \(U\) is strictly below \(H \in \mathcal{A}\) if \(U\) is below \(H\) and \(U \cap H=\varnothing\).
- Similarly, \(U\) is above or strictly above a hyperplane \(H \in \mathcal{A}\). The inversions of \(w \in W\) are the reflections that correspond to the hyperplanes \(H\) which \(w(D)\) is above.
- For a simple reflection \(s \in S\), we have \(\ell(s w)<\ell(w)\) if and only if \(s \leq w\) in the weak order if and only if \(w(D)\) is above \(H_{s}\). To decide whether \(w(D)\) is above or below \(H_{s}\) is therefore a weak order comparison.
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See Fans.
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For a Coxeter arrangement \({\mathcal{A}}\), the chambers and all their faces \(\mathcal{A}\) define the Coxeter fan \(\mathcal{F}\).
- The Coxeter fan \(\mathcal{F}\) is known to be complete, essential, and simplicial.
- The fundamental chamber \(D \in \mathcal{F}\) is a (maximal) cone spanned by the (extremal) rays \(\left\{\rho_{s} \mathrel{\Big|}s \in S\right\}\), where \(\rho_{s}\) is the intersection of \(D\) with the subspace orthogonal to the hyperplane spanned by \(\left\{\alpha_{t} \mathrel{\Big|}t \in\langle s\rangle\right\}\).
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The rays of \(\mathcal{F}\) decompose into \(n\) orbits under the action of \(W\) and each orbit contains exactly one \(\rho_{s}, s \in S\).
- Thus, any ray \(\rho \in \mathcal{F}^{(1)}\) is \(w\left(\rho_{s}\right)\) for some \(w \in W\) where \(s \in S\) is uniquely determined by \(\rho\) but \(w\) is not unique.
- In fact, \(w\left(\rho_{s}\right)=g\left(\rho_{s}\right)\) if and only if \(w \in g W_{\langle s\rangle}\).
Examples
