Norm of a Bethe vector and the Hessian of the master function

Abstract

We show that the norm of a Bethe vector in the slr+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. © Foundation Compositio Mathematica 2005.