Extending Higher Bruhat Orders to Non-longest Words in S_n

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 the symmetric group. 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