Norm of a Bethe Vector and the Hessian of the Master Function

Details Norm of a Bethe Vector and the Hessian of the Master Function Norm of a Bethe Vector and the Hessian of the Master Function

2004-02-23

arxiv.org/abs/math/0402349v1

Mathematics

Quantum Algebra

Latex, 23 pages

Scientific paper

We show that the Bethe vectors are non-zero vectors in the sl_{r+1} Gaudin model. Moreover, we show that the norm of a Bethe vector is equal to the Hessian of the corresponding master function at the corresponding non-degenerate critical point. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As other byproducts 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 and we show transversality of some Schubert cycles in the Grassmannian of r+1-dimensional planes in the space of polynomials of one variable of degree not greater than d.

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