2 00 8 Combinatorial representations of Coxeter groups over a field of two elements ∗

Let W denote a simply-laced Coxeter group with n generators. We construct an n-dimensional representation φ of W over the finite field F2 of two elements. The action of φ(W ) on F n 2 by left multiplication is corresponding to a combinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of F 2 under the action of φ(W ), and find that the kernel of φ is the center Z(W ) of W.