Non-orthogonal geometric realizations of Coxeter groups

TL;DR

This paper introduces a new axiomatic framework for Coxeter groups that generalizes classical root systems through non-orthogonal geometric realizations, enabling comparison and geometric analysis without orthogonality constraints.

Contribution

It defines a Coxeter datum for arbitrary Coxeter groups, leading to non-orthogonal root systems and new geometric insights beyond classical orthogonal representations.

Findings

Comparison between non-orthogonal and classical root systems

Development of a non-orthogonal Tits cone analysis

Establishment of geometric properties in the new framework

Abstract

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.