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Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

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Abstract

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras \({\mathfrak {g}}\), whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian \(Y({\mathfrak {g}})\). These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter Q-operators, arising from our previous study Frassek et al. (Adv. Math. 401:108283, 2022), Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional \({\mathfrak {g}}\)-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety G/P stratified by \(B_{-}\)-orbits.

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Notes

  1. The issues with \(\textrm{coker}\) not being left exact are carefully resolved by Kempf in the beginning of [Ke, §7].

  2. While the Cousin complexes were introduced by Grothendieck and were first applied in the above context in [Ke], we choose to follow the exposition of [MR] for its simplicity.

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Acknowledgements

R.F. and A.T. are indebted to Vasily Pestun for the inspiring discussions and the collaboration on [FPT], thus bringing them together close to the subject of the present note. R.F. is grateful to Gwenaël Ferrando and Volodya Kazakov for fruitful discussions. I.K. and A.T. are extremely grateful to Boris Feigin and Michael Finkelberg for the enlightening discussions of the BGG-type resolutions. A.T. is deeply grateful to Kevin Costello for a correspondence on the inspiring physics paper [CGY]; to David Hernandez for a correspondence on q-characters; to Sachin Gautam for a discussion of the Yangian’s universal R-matrices; to IHES (Bures-sur-Yvette) for the hospitality and great working conditions in July 2021 when the first stages of the present work took place. We are grateful to Zengo Tsuboi and the anonymous referee for useful suggestions and comments. R.F. received funding of the German research foundation (Deutsche Forschungsgemeinschaft DFG) Research Fellowships Programme 416527151 and support of the GNFM - INdAM. A.T. would like to gratefully acknowledge the support from NSF Grant DMS-2037602. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (QUASIFT grant agreement 677368).

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Authors and Affiliations

  1. FIM, University of Modena and Reggio Emilia, Via G. Campi 213/b, 41125, Modena, Italy

    Rouven Frassek

  2. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Ivan Karpov

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Alexander Tsymbaliuk

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  1. Rouven Frassek
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I. Karpov: On leave from National Research University Higher School of Economics, Department of Mathematics

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Frassek, R., Karpov, I. & Tsymbaliuk, A. Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions. Commun. Math. Phys. 400, 1–82 (2023). https://doi.org/10.1007/s00220-022-04620-6

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