Dualities for rational multi-particle Painlev\'e systems: Spectral versus Ruijsenaars

Abstract : The extension of the Painlev\'e-Calogero coorespondence for n-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlev\'e equations which may be obtained by the procedure of Hamiltonian reduction applied to the matrix or non-commutative Painlev\'e systems, which also gives isomonodromic formulation for these non-autonomous Hamiltonian systems. We provide here dual systems for the rational multi-particle Painlev\'e systems (PI,PII and PIV) by reduction from another intersection a coadjoint orbit of GL(n) action with the level set of moment map. We describe this duality in terms of the spectral curve of non-reduced system in comparison to the Ruijsenaars duality.
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Submitted on : Tuesday, January 28, 2020 - 9:16:28 PM
Last modification on : Tuesday, February 18, 2020 - 9:15:57 PM

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Ilia Gaiur, Vladimir Rubtsov. Dualities for rational multi-particle Painlev\'e systems: Spectral versus Ruijsenaars. 2020. ⟨hal-02458709⟩

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