Mukhin, Evgeny; Varchenko, Alexander Norm of a Bethe vector and the Hessian of the master function. (English) Zbl 1072.82012 Compos. Math. 141, No. 4, 1012-1028 (2005). Summary: We show that the norm of a Bethe vector in the \(sl_{r+1}\) Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe vectors corresponding to non-degenerate critical points are non-zero vectors. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As another byproduct of functoriality we show that the Bethe vectors form a basis in the tensor product of several copies of first and last fundamental \(sl_{r+1}\)-modules. Cited in 25 Documents MSC: 82B23 Exactly solvable models; Bethe ansatz 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 14M15 Grassmannians, Schubert varieties, flag manifolds 81R12 Groups and algebras in quantum theory and relations with integrable systems 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics Keywords:critical points; Bethe ansatz PDFBibTeX XMLCite \textit{E. Mukhin} and \textit{A. Varchenko}, Compos. Math. 141, No. 4, 1012--1028 (2005; Zbl 1072.82012) Full Text: DOI arXiv