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# Q-systems and generalizations in representation theory

#### Addabbo, Darlayne

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

## 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
- Permalink
- http://hdl.handle.net/2142/98257
- Copyright and License Information
- Copyright 2017 Darlayne Addabbo

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