Wythoff's construction for Coxeter groups

Abstract

Suppose that V is an n-dimensional real vector space, G a finite subgroup of GL( V), and P a finite subset of V. Let [G, P] be the convex cone in V generated by vectors of the form g(p), where g E G and p E P. Then [G, P] is a finitely generated cone invariant under G. If [G, P] is pointed as well, a bounded section of [G, P] by an afhne hyperplane in V is a polytope, said by Coxeter to be obtained by “Wythoff’s construction” [3]. If G, is the stabiliser of p E P in G, one might ask if the face lattice of [G, P] can be described in terms of G and its subgroups G,. This is possible under certain assumptions on V and P if G is a Coxeter group. The interest of this example lies in the fact that highly symmetrical examples can be obtained, such as the regular polytopes and many “semiregular” ones. Similar ideas for euclidean and hyperbolic Coxeter groups G lead to tesselations of those spaces [4-S]. Suppose now that W= W, is an arbitrary Coxeter group, with S finite, and Y = (T,, . . . . Tk) is a family of subsets of S. We construct in Sections l-3 an abstract “shadow lattice” L( W, Y), which is closely related to the concept of “shadows” introduced by Tits [lo]. Any interval of such a shadow lattice is again a shadow lattice, although possibly with a different number of elements in the family 9. This fact is the principal advantage of not restricting oneself to the case k = 1, as it makes inductive proofs possible. For finite W, we show that L( W, Y) is isomorphic to the face lattice of a suitable polytope [ W, P], where P = {p,, . . . . pk} is a set of points in V and W, is the stabiliser of pi in W. Alternatively, L( W, 9) can then be interpreted as a spherical tesselation. When W is euclidean or hyperbolic in the sense of [ 11, L( W, Y) is isomorphic to a tesselation of the corresponding space. 351 0021.8693/89 $3.00