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Speaker: Samuel Hsiao, University of Michigan

Title: Orienting distributive lattices

Abstract: Given a finite poset P, there is an associated distributive lattice J(P) (the "Birkhoff transform") consisting of the order ideals of P ordered by inclusion. In this talk I will introduce a signed analog of the Birkhoff transform; the notion of an order ideal will be replaced by that of a "signed filter." The resulting poset, B(P), of signed filters turns out to be Eulerian and EL-shellable; hence by a theorem of Bjorner it is the face poset of a regular CW sphere. There is a combinatorial method for computing its cd-index that has a counterpart in the flag enumerative theory of oriented matroids. In fact, B(P) seems to have the general feel of an oriented matroid, with J(P) playing the role of the geometric lattice of flats. I will also mention a close connection between chain-enumeration in B(P) and Stembridge's enriched version of the Neggers-Stanley conjecture.

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