Deformations of characters, metaplectic Whittaker functions, and the Yang-Baxter equation

Recently, unexpected connections have been discovered between characters of representations and lattice models in statistical mechanics. The bridge was first formed from Kuperberg's solution to the alternating sign matrix (ASM) conjecture. Kuperberg's proof of this conjecture, which enumerates ASMs, utilized a Yang-Baxter equation for a square ice model from statistical mechanics. In earlier work, Tokuyama and okada gave representation theoretic quantities as generating functions on certain symmetry classes of ASMs or generalizations of them. Brubaker, Bump, and Priedberg used a Yang-Baxter equation to reprove Tokuyama's result and this work seeks to do the same for a generalization of Okada's results in type B. We begin by defining the particular lattice model we study. We then imbue the lattice model with Boltzmann weights suggested by a bijection with a set of symmetric ASMs. These weights define a partition function, whose properties are studied by combinatorial and symmetric function methods over the next few chapters. This course of study culminates in the use of the Yang-Baxter equation for our ice model to prove that the partition function factors into a deformation of the Weyl denominator and a generalized character of a highest weight representation, both in type B. We conjecture that the resulting function is connected to metaplectic spherical Whittaker functions. In the last two chapters, we deal with two rather different approaches to computing Whittaker coefficients of metaplectic forms - one using a factorization of the unipotent radical to perform an integration and the other via Hecke operators on the metaplectic group.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2011

Author dc:creator

• Tabony, Sawyer James

Advisor dc:contributor.advisor

• Benjamin B. Brubaker.

• 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/1721.1/67816

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/67816