### Abstract

Q-systems and T-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain Q -functions, given as matrix elements of the action of the loop group of GL_{2} on two-component fermionic Fock space, give solutions of a Q-system. An obvious generalization using the loop group of GL_{3} acting on threecomponent fermionic Fock space leads to a new system of 4 difference equations.

Original language | English (US) |
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Article number | 023 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 15 |

DOIs | |

State | Published - Jan 1 2019 |

### Keywords

- Birkhoff factorizations
- Integrable systems
- Q- and t-systems
- τ -functions

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics
- Geometry and Topology

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## Cite this

Addabbo, D., & Bergvelt, M. (2019). τ-Functions, Birkhoff Factorizations and Difference Equations.

*Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)*,*15*, [023]. https://doi.org/10.3842/SIGMA.2019.023