Simply-laced Coxeter groups and groups generated by symplectic transvections

Boris Shapiro; Michael Shapiro; Alek Vainshtein; Andrei Zelevinsky

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arXiv · math/9906203

Simply-laced Coxeter groups and groups generated by symplectic transvections

Boris Shapiro·Michael Shapiro·Alek Vainshtein·Andrei Zelevinsky

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Abstract

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\times W. We associate to i a subgroup Gamma_i in GL_m(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 GL_m(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 Gamma_i(F_2)-orbits in F_2^m is equal to 3\times 2^s, where s is the number of simple reflections that appear in a reduced decomposition for u or v and F_2 is the two-element field.

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BibTeX RIS

Boris Shapiro, Michael Shapiro, Alek Vainshtein, Andrei Zelevinsky. 1999-06-29. Simply-laced Coxeter groups and groups generated by symplectic transvections. https://arxiv.org/abs/math/9906203

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