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

SIGMA 21 (2025), 078, 28 pages      arXiv:2412.05224      https://doi.org/10.3842/SIGMA.2025.078

Rectangular Recurrence Relations in $\mathfrak{gl}_{n}$ and $\mathfrak{o}_{2n+1}$ Invariant Integrable Models

Andrii Liashyk a, Stanislav Pakuliak b and Eric Ragoucy b
a) Beijing Institute of Mathematical Sciences and Applications (BIMSA), No. 544, Hefangkou Village Huaibei Town, Huairou District Beijing 101408, P.R. China
b) Laboratoire d'Annecy-le-Vieux de Physique Théorique (LAPTh), Chemin de Bellevue, BP 110, F-74941, Annecy-le-Vieux Cedex, France

Received February 25, 2025, in final form September 01, 2025; Published online September 21, 2025

Abstract
A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type $\mathfrak{gl}_n$ or type $\mathfrak{o}_{2n+1}$ symmetries. These recurrence relations describe how to add a single parameter $z$ to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on $z$. We refer to these recurrence relations as rectangular because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method.

Key words: Yangians; recurrence relations for Bethe vectors; nested algebraic Bethe ansatz.

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