Commutation classes of reduced words and higher Bruhat orders for affine permutations

TL;DR

This paper extends the concept of higher Bruhat orders to affine permutations, establishing a poset structure on commutation classes of reduced words and generalizing classical results.

Contribution

It introduces an analog of higher Bruhat orders for affine permutations, broadening the understanding of these structures beyond the symmetric group.

Findings

The second higher Bruhat order forms a poset on affine permutation reduced words.

Results for all higher Bruhat orders are analogous to classical cases in the symmetric group.

The construction applies to any interval in the weak order of affine permutations.

Abstract

The higher Bruhat orders are partial orders that 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. 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 both the symmetric group and the affine symmetric group. Paralleling the classical case, we show that the second higher Bruhat order is a poset on commutation classes of reduced words for any affine permutation. For the symmetric group, we also establish results for all higher Bruhat orders that are direct analogs of those in the classical case.