Simply-laced Coxeter groups and groups generated by symplectic transvections

الأصل · EN

Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W× W. We associate to i a subgroup Gammaᵢ in GLₘ(Z) generated by symplectic transvections. We prove among other things that the subgroups corresponding to different reduced words for the same pair (u,v) are conjugate to each other inside GLₘ(Z). We also generalize the enumeration result of the first three authors (see AG/9802093) by showing that, under certain assumptions on u and v, the number of Gammaᵢ(F₂)-orbits in F₂ᵐ is equal to 3× 2ˢ, where s is the number of simple reflections that appear in a reduced decomposition for u or v and F₂ is the two-element field.