Affine Higher Bruhat Orders
Abstract
The higher Bruhat orders are partial orders originally constructed by Manin and Schechtman, with an equivalent construction from Ziegler. These partial orders generalize the weak order on the symmetric group Sn, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in Sn, where covering relations correspond to braid relations. Like the weak order, each of the higher Bruhat orders is a ranked poset with unique minimal and maximal elements. Constructing analogs of the higher Bruhat orders in other settings is an area of recent interest, including recent work of Hothem for permutations in Sn. We give a construction that generalizes any interval [id,w] in the weak order where w is an element of the affine symmetric group S̃n. Our construction recovers the classical higher Bruhat orders when w is the longest element in Sn. Paralleling the classical case, we show that the second higher Bruhat order is a ranked poset on commutation classes of reduced words for w∈ S̃n, allowing us to prove a conjecture due to Elias. When w∈ Sn, we also establish results for our higher Bruhat orders that are direct analogs of numerous results in the classical higher Bruhat orders.
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BibTeX
@misc{billey2026affine,
title = {{Affine Higher Bruhat Orders}},
author = {Sara Billey and Herman Chau and Kevin Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31795},
eprint = {2610.31795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31795},
}
RIS
TY - UNPB
TI - Affine Higher Bruhat Orders
AU - Sara Billey
AU - Herman Chau
AU - Kevin Liu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31795
UR - https://arxiv.org/abs/2610.31795
AB - The higher Bruhat orders are partial orders originally constructed by Manin and Schechtman, with an equivalent construction from Ziegler. These partial orders generalize the weak order on the symmetric group S_n, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in S_n, where covering relations correspond to braid relations. Like the weak order, each of the higher Bruhat orders is a ranked poset with unique minimal and maximal elements. Constructing analogs of the higher Bruhat orders in other settings is an area of recent interest, including recent work of Hothem for permutations in S_n. We give a construction that generalizes any interval [id,w] in the weak order where w is an element of the affine symmetric group S̃_n. Our construction recovers the classical higher Bruhat orders when w is the longest element in S_n. Paralleling the classical case, we show that the second higher Bruhat order is a ranked poset on commutation classes of reduced words for w∈ S̃_n, allowing us to prove a conjecture due to Elias. When w∈ S_n, we also establish results for our higher Bruhat orders that are direct analogs of numerous results in the classical higher Bruhat orders.
ER -