# Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation

@article{Atai2021SuperMacdonaldPO, title={Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation}, author={Farrokh Atai and Martin A. Halln{\"a}s and Edwin Langmann}, journal={arXiv: Quantum Algebra}, year={2021} }

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 Macdonald-Ruijsenaars operators introduced by the same authors. We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super…

## 7 Citations

Intersecting defects and supergroup gauge theory

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2021

We consider 5d supersymmetric gauge theories with unitary groups in the Ω-background and study codim-2/4 BPS defects supported on orthogonal planes intersecting at the origin along a circle. The…

New orthogonality relations for super-Jack polynomials and an associated Lassalle--Nekrasov correspondence

- Mathematics
- 2022

. The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in n + m variables, which reduce to the Jack polynomials when n = 0 or m = 0 and provide joint…

Eigenfunctions of the van Diejen model generated by gauge and integral transformations

- Mathematics
- 2022

We present how explicit eigenfunctions of the principal Hamiltonian for the BCm relativistic Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and integral…

On refined Chern–Simons and refined ABJ matrix models

- MathematicsLetters in Mathematical Physics
- 2022

We consider the matrix model of U(N) refined Chern–Simons theory on $$S^3$$
S
3
for the unknot. We derive a q-difference operator whose insertion in the matrix integral reproduces an infinite…

Higher Order Deformed Elliptic Ruijsenaars Operators

- MathematicsCommunications in Mathematical Physics
- 2022

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…

Defects at the Intersection: the Supergroup Side

- Physics
- 2021

We consider two seemingly different theories in the Ω-background: one arises upon the most generic Higgsing of a 5d N = 1 U(N) gauge theory coupled to matter, yielding a 3d-1d intersecting defect;…

From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators

- MathematicsSelecta Mathematica
- 2021

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…

## References

SHOWING 1-10 OF 34 REFERENCES

Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials

- Mathematics
- 2007

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…

Symmetric functions and Hall polynomials

- Mathematics
- 1979

I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions…

Complete integrability of relativistic Calogero-Moser systems and elliptic function identities

- Mathematics
- 1987

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,…

Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

- Mathematics
- 2013

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…

From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators

- MathematicsSelecta Mathematica
- 2021

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…

Construction of Eigenfunctions for the Elliptic Ruijsenaars Difference Operators

- MathematicsCommunications in Mathematical Physics
- 2022

We present a perturbative construction of two kinds of eigenfunctions of the commuting family of difference operators defining the elliptic Ruijsenaars system. The first kind corresponds to elliptic…

Nonchiral intermediate long-wave equation and interedge effects in narrow quantum Hall systems

- Physics, Mathematics
- 2020

We present a nonchiral version of the intermediate long-wave (ILW) equation that can model nonlinear waves propagating on two opposite edges of a quantum Hall system, taking into account interedge…

Quantum elliptic Calogero-Moser systems from gauge origami

- Physics
- 2019

We systematically study the interesting relations between the quantum elliptic Calogero-Moser system (eCM) and its generalization, and their corresponding supersymmetric gauge theories. In…

Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type. III. Factorized Asymptotics

- MathematicsInternational Mathematics Research Notices
- 2020

In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the…