Critical points of functions, sl₂ representations, and Fuchsian differential equations with only univalued solutions

2001/12/24 by I. Scherbak, Inna Scherbak, Scherbak, I. +3 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math.CA #math.QA

paper · pdf · doi:10.48550/arxiv.math/0112269

The final version, to appear in MMJ

openalex publication_date 2001/12/24 · arxiv created 2003/01/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z₁, ..., z_n with exponents (a₁,b₁), ..., (a_n,b_n). Let the exponents at infinity be (A,B). Then for fixed generic z₁,...,z_n, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl₂ representation of dimension |A-B| in the tensor product of irreducible sl₂ representations of dimensions |a₁-b₁|, >..., |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 sl₂ KZ equation and of the Bethe vectors in the sl₂ Gaudin model. As a byproduct of this study we conclude that the Bethe vectors form a basis in the space of states for the sl₂ inhomogeneous Gaudin model.

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