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Let W be an arbitrary Coxeter group of simply-laced type (possibly infinitebut of finite rank), u,v be any two elements in W, and i be a reduced word (oflength m) for the pair (u,v) in the Coxeter group W\times W. We associate to ia subgroup Gamma_i in GL_m(Z) generated by symplectic transvections. We proveamong other things that the subgroups corresponding to different reduced wordsfor the same pair (u,v) are conjugate to each other inside GL_m(Z). We alsogeneralize the enumeration result of the first three authors (see AG/9802093)by showing that, under certain assumptions on u and v, the number ofGamma_i(F_2)-orbits in F_2^m is equal to 3\times 2^s, where s is the number ofsimple reflections that appear in a reduced decomposition for u or v and F_2 isthe two-element field.

Simply laced Coxeter groups and groups generated by symplectic transvections.