Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel

Abstract

Let $M$ be the tensor product of finite-dimensional polynomial evaluation Yangian $Y(g{l}_N)$-modules. Consider the universal difference operator $D={\sum }_{k=0}^N(-1{)}^k{T}_k(u)e^{-k{\partial }_u}$ whose coefficients $T_k(u):M\to M$ are the XXX transfer matrices associated with $M$. We show that the difference equation $Df=0$ for an $M$-valued function $f$ has a basis of solutions consisting of quasi-exponentials. We prove the same for the universal differential operator $D={\sum }_{k=0}^N(-1{)}^k{S}_k(u){\partial }_u^{N-k}$ whose coefficients $S_k(u):M\to M$ are the Gaudin transfer matrices associated with the tensor product $M$ of finite-dimensional polynomial evaluation $g{l}_N[x]$-modules.

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (3)

• Quantization of the Gaudin System
• Combinatorial Formulae for Nested Bethe Vectors
• Quasi-polynomials and the Bethe Ansatz

Cited by in corpus (5)

• KZ Characteristic Variety as the Zero Set of Classical Calogero-Moser Hamiltonians
• Solutions of ${\mathfrak{gl}}_{m|n}$ XXX Bethe ansatz equation and rational difference operators
• Lower bounds for numbers of real solutions in problems of Schubert calculus
• Reality property of discrete Wronski map with imaginary step
• Symmetric functions for the generating matrix of Yangian of gl_n(C)