Shellability of uncrossing posets and the CW poset property

Matthew Dyer’s proof that reflection orders give rise to EL-shellings for Bruhat order relied on the fact that the number of so-called ascending chains in a closed interval in Bruhat order is the leading coefficient in a polynomial closely related to the Kazhdan– Lusztig polynomial. We give a new, poset-theoretic proof that reflection orders yield EL-shellings for Bruhat order based on a characterization of cover relations in Bruhat order which may not be so well known.  This perspective also enables us to prove a conjecture of Thomas Lam that face posets of stratified spaces of response matrices of electrical networks are dual EL-shellable and are CW posets.

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