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

TL;DR

This paper connects $ au$-functions from loop group actions to solutions of $Q$-systems and introduces a new system of difference equations via generalization to ${ m GL}_3$.

Contribution

It demonstrates that certain $ au$-functions solve $Q$-systems and extends the framework to a new system of four difference equations using ${ m GL}_3$.

Findings

$ au$-functions solve $Q$-systems

Generalization to ${ m GL}_3$ yields new difference equations

Provides a group-theoretic approach to integrable difference systems

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 $\tau$-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.