Representation Theory and Spaces of Polynomials

E. Mukhin计划继续他的研究代数贝特Ansatz方法的关系，非齐次Gaudin，XXX，XXY和其他模型。 本文主要研究的问题是：发展基于Bethe Answer方程解的总体的代数Bethe Answer方法，用总体显式求解Bethe Answer方程，用总体研究Bethe Answer方程的解的个数，用总体研究Bethe Answer方程的解的个数，用总体研究Bethe Answer方程的解的个数，用总体研究Bethe Answer方程的解的个数。证明（修改后的）Bethe Answer猜想，该猜想声称Bethe向量在状态空间中形成参数的通用值的基础，在sl（N）的情况下，如果求解相应的代数方程组（称为Bethe Anchor方程组），许多数学物理模型的哈密尔顿算子就可以对角化。 证明了贝特方程的解自然地形成一族，称之为种群。 在与tosl（N）相关联的Gaudin模型中，种群与适当的Schubert圈的交点和只有多项式解的N阶标量微分算子一一对应。因此，sl（N）总体由具有指定奇点和指数的单变量多项式的N维空间给出。 E. Mukhin希望对多项式空间的研究可以使Bethe Answer方法更易于处理，并解决这一领域的一些老问题。

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.