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Séminaire Lotharingien de Combinatoire, 85B.40 (2021), 12 pp.

# Christian Gaetz and Yibo Gao

# Minimal Elements for the Limit Weak Order on Affine Weyl Groups

**Abstract.**
The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy [*Transform. Groups* **18** (2013), 179-231] in their study of total positivity for loop groups [*Adv. Math.* **230** (2012), 1222-1271]. 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
See arXiv version for a full version of this work.

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

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