Coxeter群、虚锥和优势
Coxeter groups, imaginary cones and dominance
摘要：
Brink和Howlett在Coxeter群的Tits实现中的正根上引入了一个称为优势的偏序（Math.Ann.296（1993），179- 190）。最近，将称为$\infty$ -高度的概念引入到任意Coxeter群$W$中的每个反射（Edgar，Coxeter群中的优势和规律性，博士论文，2009）。已知（Dyer，未发表），对于有限秩的所有$W$，以及对于每个非负$n$，$\infty$ -高度等于$n$的反射集是有限的。然而，尚不清楚$\infty$ -高度和优势度的概念是否相关。在这里，我们表明，$\infty$ -高度的任意反射是等于正根的数量严格占主导地位的正根对应于该反射。我们也给出了应用优势的研究虚锥的Coxeter群。
英文摘要：
Brink and Howlett have introduced a partial ordering, called dominance, on the positive roots in the Tits realization of Coxeter groups (Math. Ann. 296 (1993), 179--190). Recently a concept called $\infty$-height is introduced to each reflection in an arbitrary Coxeter group $W$ (Edgar, Dominance and regularity in Coxeter groups, PhD thesis, 2009). It is known (Dyer, unpublished) that for all $W$ of finite rank, and for each non-negative $n$, the set of reflections of $\infty$-height equal to $n$ is finite. However, it is not clear that the concepts of $\infty$-height and dominance are related. Here we show that the $\infty$-height of an arbitrary reflection is equal to the number of positive roots strictly dominated by the positive root corresponding to that reflection. We also give applications of dominance to the study of imaginary cones of Coxeter groups.
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