τ-Functions, Birkhoff Factorizations and Difference Equations

@article{Addabbo2019FunctionsBF,
  title={$\tau$-Functions, Birkhoff Factorizations and Difference Equations},
  author={Darlayne Addabbo and Maarten J. Bergvelt},
  journal={Symmetry, Integrability and Geometry: Methods and Applications},
  year={2019}
}
$Q$-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 $\tau$-functions, given as matrix elements of the action of the loop group of ${\rm GL}_{2}$ on two-component fermionic Fock space, give solutions of a $Q$-system. An obvious generalization using the loop group of ${\rm GL}_3$ acting on three-component fermionic Fock space leads… Expand

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