Shellability of uncrossing posets and the CW poset property
Connections for Women Workshop: Geometric and Topological Combinatorics August 31, 2017 - September 01, 2017
Location: MSRI: Simons Auditorium
05E15 - Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]
05C10 - Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
05A10 - Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx]
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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