Duality in elliptic Ruijsenaars system and elliptic symmetric functions

  title={Duality in elliptic Ruijsenaars system and elliptic symmetric functions},
  author={Andrei Mironov and A. Morozov and Yegor Zenkevich},
We demonstrate that the symmetric elliptic polynomials Eλ(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 yi (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, PR(x) are eigenfunctions of the elliptic reduction of the… 
Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities
We propose commuting set of matrix-valued difference operators in terms of the elliptic BaxterBelavin R-matrix in the fundamental representation of GLM . In the scalar case M = 1 these operators are
Elliptic Ruijsenaars difference operators, symmetric polynomials, and Wess-Zumino-Witten fusion rings
The fusion ring for ŝu(n)m Wess-Zumino-Witten conformal field theories is known to be isomorphic to a factor ring of the ring of symmetric polynomials presented by Schur polynomials. We introduce a
Higher Order Deformed Elliptic Ruijsenaars Operators
We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was
On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta
Abstract The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the
On Cherednik and Nazarov-Sklyanin large N limit construction for double elliptic integrable system.
The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin
Toroidal and elliptic quiver BPS algebras and beyond
Abstract The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce
Elliptic Ruijsenaars difference operators on bounded partitions
By means of a truncation condition on the parameters, the elliptic Ruijsenaars difference operators are restricted onto a finite lattice of points encoded by bounded partitions. A corresponding


Seiberg-Witten curves and double-elliptic integrable systems
A bstractAn old conjecture claims that commuting Hamiltonians of the double-elliptic integrable system are constructed from the theta-functions associated with Riemann surfaces from the
Spectral duality in integrable systems from AGT conjecture
We describe relationships between integrable systems with N degrees of freedom arising from the Alday-Gaiotto-Tachikawa conjecture. Namely, we prove the equivalence (spectral duality) between the
Complete integrability of relativistic Calogero-Moser systems and elliptic function identities
Poincaré-invariant generalizations of the Galilei-invariant Calogero-MoserN-particle systems are studied. A quantization of the classical integralsS1, ...,SN is presented such that the operatorsŜ1,
The curve of compactified 6D gauge theories and integrable systems
We analyze the Seiberg-Witten curve of the six-dimensional = (1,1) gauge theory compactified on a torus to four dimensions. The effective theory in four dimensions is a deformation of the = 2*
Shiraishi functor and non-Kerov deformation of Macdonald polynomials
We suggest a further generalization of the hypergeometric-like series due to M. Noumi and J. Shiraishi by substituting the Pochhammer symbol with a nearly arbitrary function. Moreover, this
Self-dual Hamiltonians as Deformations of Free Systems
We formulate the problem of finding self-dual Hamiltonians (associated with integrable systems) as deformations of free systems given on various symplectic manifolds and discuss several known
Integrability in String/Field Theories and Hamiltonian Flows in the Space of Physical Systems
AbstractIntegrability in string/field theories is known to emerge when considering dynamics in the moduli space of physical theories. This implies that one must study the dynamics with respect to