Extending higher Bruhat orders to non-longest words in $S_n$

Abstract: In this paper, we extend Manin and Schechtman's higher Bruhat orders for the symmetric group to higher Bruhat orders for non-longest words $w$ in $S_n$. We prove that the higher Bruhat orders of non-longest words are ranked posets with unique minimal and maximal elements. As in Manin and Schechtman's original paper, the $k$-th Bruhat order for $w$ is created out of equivalence classes of maximal chains in its $(k-1)$-st Bruhat order. We also define the second and third Bruhat orders for arbitrary realizable k-sets, and prove that the second Bruhat order has a unique minimal and maximal element. Lastly, we also outline how this extension may guide future research into developing higher Bruhat orders for affine type A Weyl groups.