Minimal elements for the limit weak order on affine Weyl groups

TL;DR

This paper classifies minimal elements in the limit weak order on all affine Weyl groups, linking them to fully commutative and Coxeter elements, and reveals their relation to Dynkin diagram nodes.

Contribution

It provides a complete classification of minimal elements across all affine types and connects these to specific classes of elements and Dynkin diagram nodes.

Findings

Infinite fully commutative elements correspond to minuscule and cominuscule nodes.

Infinite Coxeter elements relate to a single 'heavy node' in the Dynkin diagram.

Classification answers open problems in the structure of affine Weyl groups.

Abstract

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule and cominuscule nodes of the Dynkin diagram, while the infinite Coxeter elements correspond to a single node, which we call the heavy node, in all affine types other than type $A$.