Biorthogonal Laurent Polynomials,Toplitz Determinants,Minimal Toda Orbits and Isomonodromic Tau Functions |
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Authors: | M Bertola M Gekhtman |
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Institution: | (1) Department of Mathematics and Statistics, 1455 de Maisonneuve Blvd West, Montreal, Quebec H3G 1M8, Canada;(2) Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA |
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Abstract: | We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary
co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable
lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and
orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra
lattices. We establish corresponding Christoffel-Darboux formulae. For all these classes of polynomials a 2 × 2 system of
Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo-measures with arbitrary rational
logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann
sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toplitz determinants of the moments
of the pseudo-measure. In particular, the results imply that any (shifted) Toplitz (Hankel) determinant of a symbol (measure)
with arbitrary rational logarithmic derivative is an isomonodromic tau function. |
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