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

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