Duality in elliptic Ruijsenaars system and elliptic symmetric functions

Abstract

We demonstrate that the symmetric elliptic polynomials $E_{\lambda }(x)$ originally discovered in the study of generalized Noumi-Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians that act on the mother function variable $y_i$ (substitute of the Young-diagram variable $\lambda$). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $P_R(x)$ are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov (KS) Hamiltonians. This means that these latter are related to the dual eRS Hamiltonians by a somewhat mysterious orthogonality transformation, which is well defined only on the full space of time variables, while the coordinates $x_i$ appear only after the Miwa transform. This observation explains the difficulties with getting the apparently self-dual Hamiltonians from the double elliptic version of the KS Hamiltonians.

15 pages

References in corpus (5)

• Correlation functions in conformal Toda field theory II
• Seiberg-Witten curves and double-elliptic integrable systems
• Modular properties of 6d (DELL) systems
• Characteristic determinant and Manakov triple for the double elliptic integrable system
• 4d higgsed network calculus and elliptic DIM algebra

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