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$\tau$-Functions, Birkhoff Factorizations and Difference Equations
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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$Q$-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.
Keywords: integrable systems; $\tau$-functions; $Q$- and $T$-systems; Birkhoff factorizations. 
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     author = {Darlayne Addabbo and Maarten Bergvelt},
     title = {$\tau${-Functions,} {Birkhoff} {Factorizations} and {Difference} {Equations}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a22/}
}
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Darlayne Addabbo; Maarten Bergvelt. $\tau$-Functions, Birkhoff Factorizations and Difference Equations. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a22/

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