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

SIGMA 15 (2019), 023, 42 pages      arXiv:1605.00192

$\tau$-Functions, Birkhoff Factorizations and Difference Equations

Darlayne Addabbo a and Maarten Bergvelt b
a) Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
b) Department of Mathematics, University of Illinois, Urbana-Champaign, IL 61801, USA

Received July 24, 2018, in final form March 05, 2019; Published online March 27, 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 to a new system of 4 difference equations.

Key words: integrable systems; $\tau$-functions; $Q$- and $T$-systems; Birkhoff factorizations.

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