Gaudin模型与Wronski映射的generating function
Gaudin's model and the generating function of the Wronski map
摘要：
本文考虑与C^n中具有两两不同坐标的点z和不可约有限维sl 2-表示的张量积中给定权的奇异向量子空间[G]相联系的Gaudin模型.该模型的Bethe方程提供了一个显著的有理对称函数的临界点系统。任何临界轨道都决定了一个Gaudin Hamilton的公共特征向量，称为Bethe向量。在[ReV]中，证明了对于一般的z，Bethe向量跨越奇异向量的空间，即临界轨道的数量由奇异向量的空间的维度从下而上限定。相同数的上界是[SV]的主要结果之一。在本文中，我们得到这个上界在另一个，“少技术”，方法。关键的观察是，定义贝特方程的对称函数可以被解释为映射的生成函数，将一对复多项式发送到它们的朗斯基行列式中：临界轨道确定了给定多项式在此映射下的原像。在Schubert演算的框架内，临界轨道的数目可以通过特殊Schubert类的相交数来估计。与sl_2表示论[F]的关系意味着这个数是奇异向量空间的维数。我们还证明了Gaudin Hamilton算子的谱对于一般的z是简单的。
英文摘要：
We consider the Gaudin model associated to a point z in C^n with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional sl_2-representations, [G]. The Bethe equations of this model provide the critical point system of a remarkable rational symmetric function. Any critical orbit determines a common eigenvector of the Gaudin hamiltonians called a Bethe vector. In [ReV], it was shown that for generic z the Bethe vectors span the space of singular vectors, i.e. that the number of critical orbits is bounded from below by the dimension of the space of singular vectors. The upper bound by the same number is one of the main results of [SV]. In the present paper we get this upper bound in another, ``less technical'', way. The crucial observation is that the symmetric function defining the Bethe equations can be interpreted as the generating function of the map sending a pair of complex polynomials into their Wronski determinant: the critical orbits determine the preimage of a given polynomial under this map. Within the framework of the Schubert calculus, the number of critical orbits can be estimated by the intersection number of special Schubert classes. Relations to the sl_2 representation theory [F] imply that this number is the dimension of the space of singular vectors. We prove also that the spectrum of the Gaudin hamiltonians is simple for generic z.
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