1,361 research outputs found
Reality property of discrete Wronski map with imaginary step
For a set of quasi-exponentials with real exponents, we consider the discrete
Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We
prove that if the coefficients of the discrete Wronskian are real and for every
its roots the imaginary part is at most |h|, then the complex span of this set
of quasi-exponentials has a basis consisting of quasi-exponentials with real
coefficients. This result is a generalization of the statement of the B. and M.
Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe
ansatz for the XXX model.Comment: Latex, 9 page
Quasi-polynomials and the Bethe Ansatz
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 sl_2.Comment: This is the version published by Geometry & Topology Monographs on 19
March 200
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