Level-two Structure of Simply-laced Coxeter Groups

Let X be a graph, with corresponding simply-laced Coxeter group W . Then W acts naturally on the lattice L spanned by the vertices of X, preserving a quadratic form. We give conditions on X for the form to be nonsingular modulo two, and study the images of W −→ O(L/2kL). Introduction — This paper investigates the tower of 2-power congruence subgroups in a simply-laced Coxeter group, but the story begins with a puzzle for children. We have a pile of stones, and a graph X with n vertices. At most one stone may be placed on a vertex, so a vertex has one of two states: stoned or unstoned. We move by selecting a vertex v having an odd number of stoned neighbors, and then change the state of v. Given an initial configuration of stones on X, we try to reduce the total number of stones as much as possible. How to determine this minimal number of stones from the initial configuration? A configuration of stones is an element in the F2-vector space V spanned by the vertices of X. For v ∈ V , let q(v) be the number of vertices plus the number of edges in the support of v, modulo two. Then q is a quadratic form on V (see section 1), and we let O(F2) denote the subgroup of GLn(F2) preserving q. The moves are linear maps on V preserving q, and are the images of simple reflections under the natural homomorphism