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Extending higher Bruhat orders to non-longest words in S_n
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Extending higher Bruhat orders to non-longest words in S_n
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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.
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Commutation classes of reduced words and higher Bruhat orders for affine permutations
The authors define second higher Bruhat orders as posets on commutation classes of reduced words for any affine permutation and prove direct analogs of classical results for all higher orders in the symmetric group.
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