Edge labellings of partially ordered sets

Abstract

It is well known that if a finite graded lattice of rank n is supersolvable, then it has an EL-labelling where the labels along any maximal chain form a permutation of [1, 2,..., n]. We call such a labelling an Sn EL-labelling and we show that a finite graded lattice of rank n is supersolvable if and only if it has such a labelling. This result can be used to show that a graded lattice is supersolvable if and only if it has a maximal chain of left modular elements. We next study finite graded bounded posets that have Sn EL-labellings and describe a type A 0-Hecke algebra action on their maximal chains. This action is local and the resulting representation of these Hecke algebras is closely related to the flag h-vector. We show that finite graded lattices of rank n, in particular, have such an action if and only if they have an Sn EL-labelling. Our next goal is to extend these equivalences to lattices that need not be graded and, furthermore, to bounded posets that need not be lattices. In joint work with Hugh Thomas, we define left modularity in this setting, as well as a natural extension of Sn EL-labellings, known as interpolating labellings. We also suitably extend the definition of lattice supersolvability to arbitrary bounded graded posets. We show that these extended definitions preserve the appropriate equivalences. Finally, we move to the study of P-partitions. Here, edges are labelled as either "strict" or "weak" depending on an underlying labelling of the elements of the poset. A well-known conjecture of R. Stanley states that the quasisymmetric generating function for P-partitions is symmetric if and only if P is isomorphic to a Schur labelled skew shape poset.

(cont.) In characterizing these skew shape posets in terms of their local structure, C. Malvenuto made significant progress on this conjecture. We generalize the definition of P-partitions by letting the set of strict edges be arbitrary. Using cylindric diagrams, we extend Stanley's conjecture and Malvenuto's characterization to this setting. We conclude by proving both conjectures for large classes of posets.