Duality in elliptic Ruijsenaars system and elliptic symmetric functions

We demonstrate that the symmetric elliptic polynomials E λ ( 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 varia...

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Bibliographic Details
Published in:The European physical journal. C, Particles and fields Particles and fields, 2021-05, Vol.81 (5), p.1-13, Article 461
Main Authors: Mironov, A., Morozov, A., Zenkevich, Y.
Format: Article
Language:English
Subjects:
Algebra
Astronomy
Astrophysics and Cosmology
Eigenvalues
Eigenvectors
Elementary Particles
Hadrons
Hamiltonian functions
Heavy Ions
Mathematical analysis
Measurement Science and Instrumentation
Nuclear Energy
Nuclear Physics
Orthogonality
Physics
Physics and Astronomy
Polynomials
Quantum Field Theories
Quantum Field Theory
Regular Article - Theoretical Physics
String Theory
Variables
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Summary:We demonstrate that the symmetric elliptic polynomials E λ ( 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 λ ). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, P λ ( 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.
ISSN:1434-6044
1434-6052
DOI:10.1140/epjc/s10052-021-09248-9