Wythoff's construction for Coxeter groups (Q912382)

If (W,S) is a Coxeter system and T a subset of S, then a T-shadow is usually understood to be the union of \(W_ T\)-cosets \(gW_ XW_ T\) for some \(g\in W\) and \(X\subseteq S\). This concept is generalized to a family \({\mathcal F}\) of subsets of S instead of T, where an \({\mathcal F}\)-shadow is essentially a sequence \((gW_ XW_ T)_{T\in {\mathcal L}}\) of shadows parametrized by a subfamily \({\mathcal L}\) of \({\mathcal F}\). The inclusion relation among usual shadows generalizes to a natural lattice structure on the set of all \({\mathcal F}\)-shadows. The main results are that, if W is finite, Euclidean or hyperbolic, the lattice can be realized by a tesselation of a space constructed via the reflection representation of W (the space in question being a sphere if W is finite).