Representation Theory and Spaces of Polynomials

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E. Mukhin plans to continue his studies of the Algebraic Bethe Ansatzmethod in the relation to the non-homogeneous Gaudin, XXX, XXY andother models. The main problems to be addressed are: develop anapproach to the Algebraic Bethe Ansatz based on populations ofsolutions of Bethe Ansatz equations; use the populations to solve theBethe Ansatz equations explicitly; use populations to study the numberof solutions of the Bethe Ansatz equations; prove the (modified) BetheAnsatz Conjecture which claims that the Bethe vectors form a basis inthe space of states for generic values of parameters, in the case ofsl(N).Diagonalization of Hamiltonians of many models of mathematical physicscan be performed if the corresponding system of algebraic equationscalled Bethe Ansatz equations is solved. It was shown that thesolutions of the Bethe Ansatz equations naturally form families whichare called populations. In the case of the Gaudin model associated tosl(N), the populations are in one-to-one correspondence with thepoints of intersection of appropriate Schubert cycles and with thescalar differential operators of order N which have only polynomialsolutions. Thus the sl(N) populations are given by N-dimesnionalspaces of polynomials in one variable with prescribed singular points andexponents. E. Mukhin hopes that the study of the spaces ofpolynomials can make the Bethe Ansatz approach more tractable and resolveseveral old problems in this area.