Edge labellings of partially ordered sets

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2003

Author dc:creator

• McNamara, Peter, 1978-

Advisor dc:contributor.advisor

• Richard P. Stanley.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/16919

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/16919