Order types of reflection orders on affine Weyl groups

TL;DR

This paper explores the structure of reflection orders on affine Weyl groups, revealing their connection to Catalan combinatorics and providing explicit descriptions and parameterizations using deformed Dyck words.

Contribution

It explicitly characterizes all order types of reflection orders on affine Weyl groups and links them to Catalan combinatorics through deformed Dyck words.

Findings

Order types are related to Catalan combinatorics.

All order types are explicitly described.

Parameterization by deformed Dyck words is established.

Abstract

In this paper, we investigate the order types of reflection orders on irreducible affine Weyl groups. We show that they are intimately related to the Catalan combinatorics. We explicitly describe all of those order types and show that these order types can be parameterized by deformed Dyck words, which can be enumerated by Catalan numbers.