Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 060, 31 pages      math.QA/0703893      https://doi.org/10.3842/SIGMA.2007.060
Contribution to the Vadim Kuznetsov Memorial Issue

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

Evgeny Mukhin a, Vitaly Tarasov a, b and Alexander Varchenko c
a) Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA
b) St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia
c) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Received March 28, 2007; Published online April 25, 2007

Abstract
Let M be the tensor product of finite-dimensional polynomial evaluation Y(glN)-modules. Consider the universal difference operator D = åN k=0(-1)k Tk(u) e-ku whose coefficients Tk(u): M ® 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 = åNk=0(-1)kSk(u) uN-k whose coefficients Sk(u) : M® M are the Gaudin transfer matrices associated with the tensor product M of finite-dimensional polynomial evaluation glN[x]-modules.

Key words: Gaudin model; XXX model; universal differential operator.

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