Solutions to the XXX type Bethe ansatz equations and flag varieties

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We consider a version of the Aₙ Bethe equation of XXX type and introduce a reproduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the slₙ₊₁ flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems Bₙ and Cₙ. Populations of Bₙ and Cₙ type are isomorphic to the flag varieties of Cₙ and Bₙ types respectively.