Specht modules and Schubert varieties for general diagrams

The algebra of symmetric functions, the representation theory of the symmetric group, and the geometry of the Grassmannian are related to each other via Schur functions, Specht modules, and Schubert varieties, all of which are indexed by partitions and their Young diagrams. We will generalize these objects to allow for not just Young diagrams but arbitrary collections of boxes or, equally well, bipartite graphs. We will then provide evidence for a conjecture that the relation between the areas described above can be extended to these general diagrams. In particular, we will prove the conjecture for forests. Along the way, we will use a novel geometric approach to show that the dimension of the Specht module of a forest is the same as the normalized volume of its matching polytope. We will also demonstrate a new Littlewood-Richardson rule and provide combinatorial, algebraic, and geometric interpretations of it.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2010

Author dc:creator

• Liu, Ricky Ini

Advisor dc:contributor.advisor

• Alexander Postnikov.

• 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/1721.1/60196

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/60196