Séminaire Lotharingien de Combinatoire, 95B.15 (2026), 12 pp.
Sara Billey, Herman Chau, and Kevin Liu
Commutation Classes of Reduced Words and Higher Bruhat Orders for Affine Permutations
Abstract.
The higher Bruhat orders are partial orders that 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. Constructing
analogs in other settings is an area of recent interest, and we
present an analog that generalizes any interval [id,w] in the
weak order of the affine symmetric
group S~n. Paralleling the
classical case, we show the second higher Bruhat order is a poset on
commutation classes of reduced words for w
∈ S~n. When
w ∈ Sn, we also establish results
for all higher Bruhat orders that are direct analogs of ones in the
classical case.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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