Combinatorics and the KP Hierarchy
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Date
2009-10-01T15:59:46Z
Authors
Carrell, Sean
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
The study of the infinite (countable) family of partial differential equations
known as the Kadomtzev - Petviashvili (KP) hierarchy has received much interest in
the mathematical and theoretical physics community for over forty years. Recently
there has been a renewed interest in its application to enumerative combinatorics
inspired by Witten's conjecture (now Kontsevich's theorem).
In this thesis we provide a detailed development of the KP hierarchy and some of
its applications with an emphasis on the combinatorics involved. Up until now, most
of the material pertaining to the KP hierarchy has been fragmented throughout the
physics literature and any complete accounts have been for purposes much diff erent
than ours.
We begin by describing a family of related Lie algebras along with a module
on which they act. We then construct a realization of this module in terms of
polynomials and determine the corresponding Lie algebra actions. By doing this
we are able to describe one of the Lie group orbits as a family of polynomials and the
equations that de fine them as a family of partial diff erential equations. This then
becomes the KP hierarchy and its solutions. We then interpret the KP hierarchy
as a pair of operators on the ring of symmetric functions and describe their action
combinatorially. We then conclude the thesis with some combinatorial applications.
Description
Keywords
Algebraic Combintorics, Representation Theory