Affine Higher Bruhat Orders

Source 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 $\mathfrak{S}_n$, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in ${\mathfrak{S}}_n$, 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 $[\mathrm{id},w]$ in the weak order where $w$ is an element of the affine symmetric group $\widetilde{\mathfrak{S}}_n$. Our construction recovers the classical higher Bruhat orders when $w$ is the longest element in $\mathfrak{S}_n$. Paralleling the classical case, we show that the second higher Bruhat order is a ranked poset on commutation classes of reduced words for $w\in \widetilde{\mathfrak{S}}_n$, allowing us to prove a conjecture due to Elias. When $w\in \mathfrak{S}_n$, we also establish results for our higher Bruhat orders that are direct analogs of numerous results in the classical higher Bruhat orders.