We study solutions of the Bethe Ansatz equation related to the
trigonometric Gaudin model associated to a simple Lie algebra g
and a tensor product of irreducible finite-dimensional
representations. Having one solution, we describe a construction of
new solutions. The collection of all solutions obtained from a
given one is called a population. We show that the Weyl group of
g acts on the points of a population freely and transitively (under
certain conditions).
To a solution of the Bethe Ansatz equation, one assigns a common
eigenvector (called the Bethe vector) of the trigonometric Gaudin
operators. The dynamical Weyl group projectively acts on the common
eigenvectors of the trigonometric Gaudin operators. We conjecture
that this action preserves the set of Bethe vectors and
coincides with the action induced by the action on points of
populations. We prove the conjecture for sl2.
Keywords
Bethe Ansatz, trigonometric Gaudin model,
XXX model