Cone Types, Automata, and Regular Partitions in Coxeter Groups

TL;DR

This paper introduces regular partitions in Coxeter groups, linking them to automata recognizing reduced words, and proves the existence of unique minimal length representatives for each cone type, advancing the understanding of Coxeter group structure.

Contribution

It develops the theory of regular partitions, establishes their equivalence to automata recognizing reduced words, and proves the uniqueness of minimal length representatives for cone types.

Findings

Regular partitions are equivalent to automata recognizing reduced words.

Each cone type has a unique minimal length element.

Boundary roots characterize cone types as intersections of half-spaces.

Abstract

In this article we introduce the notion of a \textit{regular partition} of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group. As an application of this theory we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi's classical result that each component of the Shi arrangement of an affine Coxeter group has a unique minimal length element. We further develop the theory of cone types in Coxeter groups by identifying the minimal set of roots required to express a cone type as an intersection of half-spaces. This set of \textit{boundary roots} is closely related to the elementary inversion sets of Brink and Howlett, and also to the notion of the base of an inversion set introduced by Dyer.