Elliptic Ruijsenaars-Schneider integrable systems and elliptic Macdonald polynomials

Yegor Zenkevich ( SISSA )

Tuesday 7th February 16:00-17:00 Maths 311B


We introduce an elliptic generalization of Macdonald symmetric
functions with the following remarkable properties:

1) These polynomials are eigenfunctions of the elliptic
Ruijsenaars-Scheneider Hamiltonians acting on their indices.

2) The structure constants of the ring of elliptic Macdonald
polynomials are natural elliptizations of the (q,t)-deformed
Littlewood-Richardson coefficients, and most importantly they vanish
whenever standard Littlewood-Richardson coefficients vanish.

We describe the (bi)spectral dual of the Ruijsenaars-Schenider
Hamiltonians, i.e. the Hamiltonians acting on the variables in
elliptic Macdonald polynomials, which turn out to be related to the
Hamiltonians of the double-elliptic integrable system recently
proposed by Koroteev and Shakirov. As an application of our findings
we obtain a new description of certain representation of elliptic
quantum toroidal algebra of type gl(1).

The talk is based on the joint paper with A. Morozov and A. Mironov

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