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

SIGMA 15 (2019), 023, 42 pages      arXiv:1605.00192      https://doi.org/10.3842/SIGMA.2019.023

### $\tau$-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

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.

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

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References

1. Addabbo D., Bergvelt M., Generalizations of $Q$-systems and orthogonal polynomials from representation theory, in Lie algebras, Vertex Operator Algebras, and Related Topics, Contemp. Math., Vol. 695, Amer. Math. Soc., Providence, RI, 2017, 1-13, arXiv:1604.02190.
2. Addabbo D., Bergvelt M., Difference hierarchies for $nT$ $\tau$-functions, Internat. J. Math. 29 (2018), 1850090, 29 pages, arXiv:1611.10340.
3. Adler M., On the Bäcklund transformation for the Gel'fand-Dickey equations, Comm. Math. Phys. 80 (1981), 517-527.
4. Alexandrov A., Zabrodin A., Free fermions and tau-functions, J. Geom. Phys. 67 (2013), 37-80, arXiv:1212.6049.
5. Barrios Rolanía D., Branquinho A., Foulquié Moreno A., On the relation between the full Kostant-Toda lattice and multiple orthogonal polynomials, J. Math. Anal. Appl. 377 (2011), 228-238, arXiv:0911.2856.
6. Bergvelt M.J., ten Kroode A.P.E., $\tau$ functions and zero curvature equations of Toda-AKNS type, J. Math. Phys. 29 (1988), 1308-1320.
7. Bergvelt M.J., ten Kroode A.P.E., Partitions, vertex operator constructions and multi-component KP equations, Pacific J. Math. 171 (1995), 23-88, arXiv:hep-th/9212087.
8. Borcherds R.E., Vertex algebras, in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math., Vol. 160, Birkhäuser Boston, Boston, MA, 1998, 35-77.
9. Bressoud D.M., Proofs and confirmations. The story of the alternating sign matrix conjecture, MAA Spectrum, Cambridge University Press, Cambridge, 1999.
10. Date E., Kashiwara M., Jimbo M., Miwa T., Transformation groups for soliton equations, in Nonlinear Integrable Systems - Classical Theory and Quantum Theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, 39-119.
11. Di Francesco P., Kedem R., Positivity of the $T$-system cluster algebra, Electron. J. Combin. 16 (2009), 140, 39 pages, arXiv:0908.3122.
12. Di Francesco P., Kedem R., $Q$-systems, heaps, paths and cluster positivity, Comm. Math. Phys. 293 (2010), 727-802, arXiv:0811.3027.
13. Frenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Inc., Boston, MA, 1988.
14. Frenkel I.B., Huang Y.-Z., Lepowsky J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), viii+64 pages.
15. Hernandez D., The Kirillov-Reshetikhin conjecture and solutions of $T$-systems, J. Reine Angew. Math. 596 (2006), 63-87, arXiv:math.QA/0501202.
16. Hirota R., The direct method in soliton theory, Cambridge Tracts in Mathematics, Vol. 155, Cambridge University Press, Cambridge, 2004.
17. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
18. Kac V.G., Vertex algebras for beginners, 2nd ed., University Lecture Series, Vol. 10, Amer. Math. Soc., Providence, RI, 1998.
19. Kac V.G., Peterson D.H., Lectures on the infinite wedge-representation and the MKP hierarchy, in Systèmes dynamiques non linéaires: intégrabilité et comportement qualitatif, Sém. Math. Sup., Vol. 102, Presses University Montréal, Montreal, QC, 1986, 141-184.
20. Kac V.G., Raina A.K., Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, Vol. 2, World Sci. Publ. Co., Inc., Teaneck, NJ, 1987.
21. Kasman A., Orthogonal polynomials and the finite Toda lattice, J. Math. Phys. 38 (1997), 247-254.
22. Kirillov A.N., Reshetikhin N.Yu., Representations of Yangians and multiplicities of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras, J. Soviet Math. 52 (1990), 3156-3164.
23. Kirillov A.N., Reshetikhin N.Yu., Formulas for multiplicities of occurence of irreducible components in the tensor product of representations of simple Lie algebras, J. Soviet Math. 80 (1996), 1768-1772.
24. Kuniba A., Nakanishi T., Suzuki J., $T$-systems and $Y$-systems in integrable systems, J. Phys. A: Math. Theor. 44 (2011), 103001, 146 pages, arXiv:1010.1344.
25. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
26. Miwa T., Jimbo M., Date E., Solitons. Differential equations, symmetries and infinite-dimensional algebras, Cambridge Tracts in Mathematics, Vol. 135, Cambridge University Press, Cambridge, 2000.
27. Peterson D.H., Kac V.G., Infinite flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci. USA 80 (1983), 1778-1782.
28. Pressley A., Segal G., Loop groups, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1986.
29. Segal G., Wilson G., Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5-65.
30. Sklyanin E.K., Bäcklund transformations and Baxter's $Q$-operator, in Integrable Systems: from Classical to Quantum (Montréal, QC, 1999), CRM Proc. Lecture Notes, Vol. 26, Amer. Math. Soc., Providence, RI, 2000, 227-250, arXiv:nlin.SI/0009009.
31. ten Kroode F., van de Leur J., Bosonic and fermionic realizations of the affine algebra $\widehat{\rm gl}_n$, Comm. Math. Phys. 137 (1991), 67-107.
32. Wilson G., Habillage et fonctions $\tau$, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 587-590.