On the Weak Order of Orthogonal Groups

Abstract

A complete lattice structure is defined on the underlying set of the orthogonal group of a real Euclidean space, by a construction analogous to that of the weak order of Coxeter systems in terms of root systems. This produces a complete rootoid in the sense of Dyer, with the orthogonal group as underlying group. It is shown that this complete lattice has a saturation property which is used along with other properties of the lattice to characterize the maximal totally ordered subsets of the lattice as collections of initial sections with respect to a total ordering on the positive roots.

Key Words:

• Convex cone
• Groupoid
• Lattice
• Orthogonal group
• Root system
• Weak order

2010 Mathematics Subject Classification:

• 06B23
• 20L05
• 17B22
• 20F55

ACKNOWLEDGMENT

I would like to thank Matthew Dyer for some very helpful conversations.

Notes

Communicated by M. Kambites.