Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials

@article{Sergeev2007DeformedMO,
  title={Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials},
  author={Alexander N. Sergeev and A P Veselov},
  journal={Communications in Mathematical Physics},
  year={2007},
  volume={288},
  pages={653-675}
}
It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald- Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced; the combinatorial formulas for them are given. 
Macdonald–Koornwinder Polynomials
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases.
Generalized Macdonald-Ruijsenaars systems
Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation
The super-Macdonald polynomials, introduced by Sergeev and Veselov, generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed
Cherednik operators and Ruijsenaars-Schneider model at infinity
Heckman introduced N operators on the space of polynomials in N variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack
From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators
Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By specialisations of his
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
Lie Superalgebras and Calogero–Moser–Sutherland Systems
  • A. Sergeev
  • Mathematics
    Journal of Mathematical Sciences
  • 2018
We review recent results obtained at the intersection of the theory of quantum deformed Calogero–Moser–Sutherland systems and the theory of Lie superalgebras. We begin with a definition of admissible
Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models
We consider the relativistic generalization of the quantum AN-1 Calogero–Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these
Orbits and invariants of super Weyl groupoid
We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra $\mathfrak {gl}(n,m)$, depending on a parameter. We show that for generic values
On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums
AbstractGeneralized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such
...
...

References

SHOWING 1-10 OF 36 REFERENCES
Deformed Quantum Calogero-Moser Problems and Lie Superalgebras
The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized
(Shifted) Macdonald polynomials: q-Integral representation and combinatorial formula
We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for
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,
Interpolation, Integrality, and a Generalization of Macdonald's Polynomials
In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values 1/2, 1, and 2 they
Coincident root loci and Jack and Macdonald polynomials for special values of the parameters
We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms
Difference equations and symmetric polynomials defined by their zeros
In this paper, we are starting a systematic analysis of a class of symmetric polynomials which, in full generality, has been introduced in [Sa]. The main features of these functions are that they are
Symmetric and non-symmetric quantum Capelli polynomials
Generalizing the classical Capelli identity has recently attracted a lot of interest ([HU], [Ok], [Ol], [Sa], [WUN]). In several of these papers it was realized, in various degrees of generality,
...
...