### References & Citations

# High Energy Physics - Theory

# Title: Duality in elliptic Ruijsenaars system and elliptic symmetric functions

(Submitted on 3 Mar 2021 (v1), last revised 4 Mar 2021 (this version, v2))

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.

## Submission history

From: Andrei Mironov [view email]**[v1]**Wed, 3 Mar 2021 16:27:42 GMT (72kb,D)

**[v2]**Thu, 4 Mar 2021 16:50:05 GMT (72kb,D)

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