Coxeter groups, imaginary cones and dominance

Details Coxeter groups, imaginary cones and dominance Coxeter groups, imaginary cones and dominance

2011-08-26

arxiv.org/abs/1108.5232v2

Mathematics

Representation Theory

Scientific paper

Brink and Howlett have introduced a partial ordering, called \emph{dominance}, on the positive roots in the Tits realization of Coxeter groups (Math. Ann. \textbf{296} (1993), 179--190). Recently a concept called \emph{$\infty$-height} is introduced to each reflection in an arbitrary Coxeter group $W$ (Edgar, \emph{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 It is known (Dyer, unpublished) that for all $W$ of finite rank, and for each non-negative $n$, the set of reflections of $\infty$-height equal to $n$ is finite.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 \emph{imaginary cones} of Coxeter groups.

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