Imaginary cone and reflection subgroups of Coxeter groups
Imaginary cone and reflection subgroups of Coxeter groups
复制标题
Coxeter 群的假想锥体和反射子群
作者：
M. Dyer
Kac-Moody李代数的虚锥是零的凸壳和正虚根。本文研究了一般Coxeter群W的一类根系的虚锥，证明了W的一个反射子群的虚锥包含在W的虚锥中，并且对于有限秩不可约无限的W，闭虚锥是单根所跨的点锥中唯一包含的非零的闭的点W-稳定锥.对于有限秩W，证明了虚锥的面的各种自然概念是重合的，面格是用面反射子群的格来刻画的，并且证明了Tits锥和虚锥是由类似于多面体锥的标准对偶的对偶而联系的，尽管它们通常都不是闭锥.其中一些结果可应用于一般Iwahori-Hecke代数的Coxeter群的占优序、伴随自动机和模的构造，这些结果将在本文的续文中给出。
The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W, and that for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero, closed, pointed W-stable cone contained in the pointed cone spanned by the simple roots. For W of finite rank, various natural notions of faces of the imaginary cone are shown to coincide, the face lattice is explicitly described in terms of the lattice of facial reflection subgroups and it is shown that the Tits cone and imaginary cone are related by a duality closely analogous to the standard duality for polyhedral cones, even though neither of them is a closed cone in general. Some of these results have application, to be given in sequels to this paper, to dominance order of Coxeter groups, associated automata, and construction of modules for generic Iwahori-Hecke algebras.