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

We show that the Bethe vectors are non-zero vectors in the slr+1 Gaudin model. Namely, 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 slr+1 modules and we show transversality of some Schubert cycles in the Grassmannian of r + 1-dimensional planes in the space Cd[x] of polynomials of one variable of degree not greater than d. ∗ Department of Mathematical Sciences, Indiana University Purdue University Indianapolis, 402 North Blackford St., Indianapolis, IN 46202-3216, USA Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA