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

TL;DR

This paper generalizes higher Bruhat orders from the longest words in the symmetric group to arbitrary words, establishing their structure as ranked posets and exploring their properties for realizable k-sets.

Contribution

It extends the concept of higher Bruhat orders to non-longest words in $S_n$, proving their poset structure and properties, and suggests directions for affine type A Weyl groups.

Findings

Higher Bruhat orders for non-longest words are ranked posets with unique minimal and maximal elements.

The second Bruhat order for realizable k-sets has a unique minimal and maximal element.

Outline of potential extensions to affine type A Weyl groups.

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.