Infinite Reduced Words, Lattice Property And Braid Graph of Affine Weyl Groups

TL;DR

This paper explores the structure of infinite reduced words in affine Weyl groups, establishing a bijection with biclosed sets, and proves the connectedness of their braid graphs, extending classical results to an infinite setting.

Contribution

It introduces a bijection between infinite reduced words and biclosed sets, and proves the lattice and connectivity properties of affine Weyl groups' braid graphs.

Findings

Bijection between infinite reduced words and biclosed sets.

Biclosed sets form a complete algebraic ortholattice in rank 3 affine Weyl groups.

Braid graphs of these groups are connected, extending finite case results.

Abstract

In this paper, we establish a bijection between the infinite reduced words of an affine Weyl group and certain biclosed sets of its positive system and determine all finitely generated biclosed sets in the positive system of an affine Weyl group. Using these results, we show first that the biclosed sets in the standard positive system of rank 3 affine Weyl groups when ordered by inclusion form a complete algebraic ortholattice and secondly that the (generalized) braid graphs of those Coxeter groups are connected, which can be thought of as an infinite version of Tit's solution to the word problem.