τ-Functions, Birkhoff Factorizations and Difference Equations

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 GL₂ on two-component fermionic Fock space, give solutions of a Q-system. An obvious generalization using the loop group of GL₃ acting on threecomponent fermionic Fock space leads to a new system of 4 difference equations.