Combinatorial representations of Coxeter groups over a field of two elements

Abstract

Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $ϕ$ of $W$ over the finite field $F_2$ of two elements. The action of $ϕ(W)$ on $F_2^n$ 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 $F_2^n$ under the action of $ϕ(W)$, and find that the kernel of $ϕ$ is the center $Z(W)$ of $W.$

20 pages, 2 figures, 2 tables