Combinatorial representations of Coxeter groups over a field of two elements

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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 F₂ⁿ 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 F₂ⁿ under the action of ϕ(W), and find that the kernel of ϕ is the center Z(W) of W.