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The order dimension of the poset of regions in a hyperplane arrangement. (English) Zbl 1044.52010

The order dimension of a finite poset \(P\) is the smallest \(n\) so that \(P\) can be embedded as an induced subposet of the componentwise order on \(\mathbb{R}^n\). In this paper, the author shows that the order dimension of the weak order on a Coxeter group of type \({\mathbf A}\), \({\mathbf B}\) or \({\mathbf C}\) (that is, the order dimension of the poset of regions of the corresponding Coxeter arrangement) is equal to the rank of the Coxeter group. The case for \({\mathbf A}_n\) was proven previously in [S. Flath, Order 10, No. 3, 201–219 (1993; Zbl 0795.06004)] (see also J. L. Ramírez Alfonsín and D. Romero, Discrete Math. 254, 473–483 (2002; Zbl 1027.05029) where explicit embeddings are given).

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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