Critical points of functions, sl_2 representations, and Fuchsian differential equations with only univalued solutions
Critical points of functions, sl_2 representations, and Fuchsian differential equations with only univalued solutions
复制标题
函数、sl_2 表示和仅具有单值解的 Fuchsian 微分方程的临界点
DOI：
10.17323/1609-4514-2003-3-2-621-645
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
A. Varchenko
中科院分区：
文献类型：
--
作者：
I. Scherbak;A. Varchenko
Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z_1, ..., z_n with exponents (a_1,b_1), ..., (a_n,b_n). Let the exponents at infinity be (A,B). Then for fixed generic z_1,...,z_n, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl_2 representation of dimension |A-B| in the tensor product of irreducible sl_2 representations of dimensions |a_1-b_1|, >..., |a_n-b_n|.
To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hypergeometric solutions of the sl_2 KZ equation and of the Bethe vectors in the sl_2 Gaudin model. As a byproduct of this study we conclude that the Bethe vectors form a basis in the space of states for the sl_2 inhomogeneous Gaudin model.
