On The Weak Order Of Orthogonal Groups

TL;DR

This paper introduces a lattice structure on the orthogonal group of a Euclidean space, analogous to the weak order in Coxeter systems, leading to a new algebraic framework called a complete rootoid.

Contribution

It constructs a complete lattice (weak order) on orthogonal groups, extending Coxeter system concepts to a broader geometric setting.

Findings

Defines a complete lattice structure on orthogonal groups.

Establishes the orthogonal group as a complete rootoid.

Provides a new algebraic perspective on orthogonal groups.

Abstract

A structure of a complete lattice (in the sense of a poset) is defined on the underlying set of the orhtogonal group of a real Euclidean space, by a construction analogous to that of the weak order of a Coxeter system in terms of its root system. This gives rise to a complte rootoid in the sense of Dyer, with the orthogonal group as underlying group.