%0 Journal Article
%T Non-orthogonal geometric realizations of Coxeter groups
%A Xiang Fu
%J Mathematics
%D 2011
%I arXiv
%R 10.1017/S1446788714000263
%X We define in an axiomatic fashion a \emph{Coxeter datum} for an arbitrary Coxeter group $W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear paring without any integrality and non-degeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root systems of Coxeter groups. We obtain comparison results between these non-orthogonal root systems and the classical root systems. Further, we study the equivalent of the Tits cone in these non-orthogonal representations, and we show that strong results on the geometry in the equivalent of the Tits cone can be obtained.
%U http://arxiv.org/abs/1112.3429v5