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

SIGMA 21 (2025), 060, 28 pages      arXiv:2312.00980      https://doi.org/10.3842/SIGMA.2025.060

New Combinatorial Formulae for Nested Bethe Vectors

Maksim Kosmakov a and Vitaly Tarasov b
a) Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH 45221, USA
b) Department of Mathematical Sciences, Indiana University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA

Received January 08, 2025, in final form July 08, 2025; Published online July 22, 2025

Abstract
We give new combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for the evaluation modules over the Yangian $Y(\mathfrak{gl}_4)$. The case of $Y(\mathfrak{gl}_n)$ for an arbitrary $n$ is considered in [Lett. Math. Phys. 115 (2025), 12, 20 pages, arXiv:2402.15717].

Key words: Bethe ansatz; Yangian; weight functions.

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References

  1. Felder G., Rimányi R., Varchenko A., Poincaré-Birkhoff-Witt expansions of the canonical elliptic differential form, in Quantum Groups, Contemp. Math., Vol. 433, American Mathematical Society, Providence, RI, 2007, 191-208, arXiv:math.RT/0502296.
  2. Khoroshkin S., Pakuliak S., Generating series for nested Bethe vectors, SIGMA 4 (2008), 081, 23 pages, arXiv:0810.3131.
  3. Khoroshkin S., Pakuliak S., Tarasov V., Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57 (2007), 1713-1732, arXiv:math.QA/0610517.
  4. Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
  5. Kosmakov M., Tarasov V., New combinatorial formulae for nested Bethe vectors II, Lett. Math. Phys. 115 (2025), 12, 20 pages, arXiv:2402.15717.
  6. Kulish P.P., Reshetikhin N.Yu., Diagonalisation of ${\rm GL}(N)$ invariant transfer matrices and quantum $N$-wave system (Lee model), J. Phys. A 16 (1983), L591-L596.
  7. Kulish P.P., Reshetikhin N.Yu., ${\rm GL}_{3}$-invariant solutions of the Yang-Baxter equation and associated quantum systems, J. Sov. Math. 34 (1986), 1948-1971.
  8. Markov Y., Varchenko A., Hypergeometric solutions of trigonometric Knizhnik-Zamolodchikov equations satisfy dynamical difference equations, Adv. Math. 166 (2002), 100-147.
  9. Matsuo A., An application of Aomoto-Gel'fand hypergeometric functions to the ${\rm SU}(n)$ Knizhnik-Zamolodchikov equation, Comm. Math. Phys. 134 (1990), 65-77.
  10. Miwa T., Takeyama Y., Tarasov V., Determinant formula for solutions of the quantum Knizhnik-Zamolodchikov equation associated with $U_q({\rm sl}_n)$ at $|q|=1$, Publ. Res. Inst. Math. Sci. 35 (1999), 871-892, arXiv:math.QA/9812096.
  11. Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 2006 (2006), P08002, 44 pages, arXiv:math.QA/0605015.
  12. Mukhin E., Tarasov V., Varchenko A., Spaces of quasi-exponentials and representations of the Yangian $Y(\mathfrak{gl}_N)$, Transform. Groups 19 (2014), 861-885, arXiv:1303.1578.
  13. Oskin A., Pakuliak S., Silantyev A., On the universal weight function for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, St. Petersburg Math. J. 21 (2009), 651-680, arXiv:0711.2821.
  14. Rimányi R., Stevens L., Varchenko A., Combinatorics of rational functions and Poincaré-Birchoff-Witt expansions of the canonical $U(\mathfrak{n}_-)$-valued differential form, Ann. Comb. 9 (2005), 57-74, arXiv:math.CO/0407101.
  15. Rimányi R., Tarasov V., Varchenko A., Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions, Math. Z. 277 (2014), 1085-1104, arXiv:1204.4961.
  16. Rimányi R., Tarasov V., Varchenko A., Partial flag varieties, stable envelopes, and weight functions, Quantum Topol. 6 (2015), 333-364, arXiv:1212.6240.
  17. Rimányi R., Tarasov V., Varchenko A., Trigonometric weight functions as $K$-theoretic stable envelope maps for the cotangent bundle of a flag variety, J. Geom. Phys. 94 (2015), 81-119, arXiv:1411.0478.
  18. Rimányi R., Tarasov V., Varchenko A., Elliptic and $K$-theoretic stable envelopes and Newton polytopes, Selecta Math. (N.S.) 25 (2019), 16, 43 pages, arXiv:1705.09344.
  19. Schechtman V., Varchenko A., Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990), 279-283.
  20. Schechtman V., Varchenko A., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194.
  21. Slavnov N., Algebraic Bethe ansatz, arXiv:1804.07350.
  22. Slavnov N., Algebraic Bethe ansatz and correlation functions—an advanced course, World Scientific Publishing, Hackensack, NJ, 2022.
  23. Takhtadzhan L.A., Faddeev L.D., The quantum method for the inverse problem and the $XYZ$ Heisenberg model, Russ. Math. Surv. 34 (1979), 11-68.
  24. Tarasov V., Varchenko A., Selberg-type integrals associated with $\mathfrak{sl}_3$, Lett. Math. Phys. 65 (2003), 173-185, arXiv:math.QA/0302148.
  25. Tarasov V., Varchenko A., Combinatorial formulae for nested Bethe vectors, SIGMA 9 (2013), 048, 28 pages, arXiv:math.QA/0702277.
  26. Tarasov V., Varchenko A., Hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety, Cent. Eur. J. Math. 12 (2014), 694-710, arXiv:1301.2705.
  27. Tarasov V., Varchenko A., $q$-hypergeometric solutions of quantum differential equations, quantum Pieri rules, and gamma theorem, J. Geom. Phys. 142 (2019), 179-212, arXiv:1710.03177.
  28. Varchenko A., Tarasov V., Jackson integral representations for solutions of the Knizhnik-Zamolodchikov quantum equation, St. Petersburg Math. J. 6 (1994), 275-313, arXiv:hep-th/9311040.

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