Q-systems and generalizations in representation theory

Addabbo, Darlayne

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https://hdl.handle.net/2142/98257 Copy

Description

Title

Q-systems and generalizations in representation theory

Author(s)

Addabbo, Darlayne

Issue Date

2017-07-10

Director of Research (if dissertation) or Advisor (if thesis)

Bergvelt, Maarten

Doctoral Committee Chair(s)

Kedem, Rinat

Committee Member(s)

• Di Francesco, Philippe
• Nevins, Thomas

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Q-systems
• Representation theory
• Integrable systems
• Box and ball systems

Abstract

We study tau-functions given as matrix elements for the action of loop groups, $\widehat{GL_n}$ on $n$-component fermionic Fock space. In the simplest case, $n=2$, the tau-functions are equal to Hankel determinants and applying the famous Desnanot-Jacobi identity, one can see that they satisfy a $Q$-system. Since $Q$-systems are of interest in many areas of mathematics, it is interesting to study tau-functions and the discrete equations they satisfy for the $n>2$ cases. We generalize this work by studying tau-functions equal to matrix elements for the action of infinite matrix groups, denoted $\widehat{GL}_{\infty}^{(n)}$ on $n$-component fermionic Fock space. The $n=2$ case, similarly to the $\widehat{GL_2}$ situation, gives tau-functions which have a simple determinantal formula and the relations they satisfy are again obtained by applying the Desnanot-Jacobi identity. In this case, the tau-functions satisfy $T$-system relations. In the following, we will define our tau-functions and explain how to compute them and then present multiple ways of deriving the relations that they satisfy, which is much more complicated in the $n>2$ cases. The method of ultra-discretization provides a way to obtain from discrete integrable equations, combinatorial models that maintain the essential properties of the original equations. With some extra initial conditions, the $Q$-system for our $\widehat{GL_2}$ case is also known as the discrete finite $1$-dimensional Toda molecule equation. It is known that this can be ultra-discretized to obtain the famous Box and Ball system. In the final chapter of this thesis, we present a new generalization of the Box and Ball system obtained by ultra-discretizing the $T$-system (discrete finite $2$-dimensional Toda molecule equation).

Graduation Semester

2017-08

Type of Resource

text

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http://hdl.handle.net/2142/98257

Copyright and License Information

Copyright 2017 Darlayne Addabbo

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