Q-systems and generalizations in representation theory

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{GL₂} 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{GL₂} 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).

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Addabbo, Darlayne

Contributors dc:contributor

• Bergvelt, Maarten
• Kedem, Rinat
• Di Francesco, Philippe
• Nevins, Thomas

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

Statement dc:rights

• Copyright 2017 Darlayne Addabbo

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/98257