What's concrete model for Coxeter complexes?

Question

We know for every Coxeter system $(W,S)$ there is a Coxeter complex associated by its cosets of parabolic subgroups. In Wachs's note Poset Topology p.12-13 she mentioned for the Coxeter complex of type A, it is just the order complex of the Boolean lattice which is the lattice of subsets of $[n]$. And she also mentioned that the Coxeter complex of type B is the order complexes of the dual of face lattices of $n$-cube. So I think there should exist concrete models for other types of Coxeter groups like type A and B, but I couldn't find similar results. Does someone know it?

Answer 1

Let me record the answer from the comments here.

Suppose $(W,S)$ is a finite Coxeter system, with associated root system $\Phi$ in Euclidean vector space $V$. (I don't think the assumption of crystallography is needed in anything I'm about to say, but if you care specifically about Type D, that is a crystallographic type so things will be even better there.)

The Coxeter arrangement associated to $\Phi$ is the central hyperplane arrangement with hyperplanes orthogonal to the roots. The Coxeter complex $\Sigma$ of $W$ can be realized by instersecting the Coxeter arrangement with a small sphere around the origin, so the chambers of the hyperplane arrangement become the facets of the complex, the rays of the arrangement become the vertices of the complex, and so on. This is a flag simplicial complex, homeomorphic to a sphere. Note in particular that its facets correspond to the elements of $W$. A nice reference for this construction is Section 11.6 of T. Kyle Petersen's "Eulerian Numbers" textbook.

As mentioned by the question-asker, in Types A and B the Coxeter arrangement is an order complex of a poset: in Type A it is the order complex of the Boolean lattice, a.k.a. face lattice of the simplex; in Type B we use the face lattice of the hypercube instead. But it is not always the case that the Coxeter complex is an order complex of a poset. For instance, in Type D this is not the case. In fact, I believe the Coxeter complex is an order complex of a poset exactly when the Coxeter diagram of $\Phi$ has no "forks" (i.e., is a path, when ignoring edge decorations). These are exactly the Coxeter diagrams that correspond to regular polytopes and then I believe the Coxeter complex will be the order complex of the face lattice of the corresponding regular polytope.

The notion of Coxeter complex applies to infinite Coxeter groups as well, but I will not explain this; see Wikipedia.

For finite ("spherical") Coxeter groups, the Coxeter complex as mentioned is homeomorphic to a sphere; but even better, it is polytopal. The permutohedron of $W$ (sometimes called "Coxterhedron" in earlier references) is the convex hull of the $W$ orbit of any point in $V$ not on any hyperplane in the Coxeter arrangement. Its combinatorial structure does not depend on which point in the fundamental chamber we choose. The $W$-permutohedron is a simple polytope, and its vertices are in bijection with the elements of $W$.

In fact, there is another way to realize the $W$-permutohedron as a zonotope: it is the Minkowksi sum $\sum_{\alpha \in \Phi^+}[-\alpha/2,\alpha/2]$ (here the notation $[-\alpha/2,\alpha/2]$ means the line segment joining $-\alpha/2$ to $\alpha/2$; you can just think of this as the Minkowski sum of the roots of $\Phi$).

The permutohedron and the Coxeter complex are basically the same thing: precisely, the Coxeter complex is the dual of the permutohedron (the dual of a simple polytope is a simplicial polytope). See for instance the article "Coxeter-associahedra" of Reiner and Ziegler mentioned above for a quick review of these facts.

This polytopal construction gives a rather concrete model for the Coxeter complex in all the finite types. You can work out the combinatorics of the Type D case specifically from this construction, but there are also probably places where it is spelled out already.