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Macdonald polynomials and algebraic integrability

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arxiv math/0212313 v1 pith:UEC4SIOC submitted 2002-12-22 math.QA math.CO

Oleg Chalykh This is my paper · ORCID
classification math.QAmath.CO
keywords macdonaldalgebraiccaseintegrabilitymathbboperatorspolynomialsroot
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abstract

We construct explicitly non-polynomial eigenfunctions of the difference operators by Macdonald in case $t=q^k$, $k\in{\mathbb Z}$. This leads to a new, more elementary proof of several Macdonald conjectures, first proved by Cherednik. We also establish the algebraic integrability of Macdonald operators at $t=q^k$ ($k\in {\mathbb Z}$), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including $BC_n$ case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of $A_n$ root system where the previously known methods do not work.

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Cited by 8 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Cherednik integrable system: eigenfunctions at generic eigenvalues

    hep-th 2026-07 conditional novelty 7.0 of 10

    Generic Cherednik eigenfunctions are N!-branched power series obtained by analytic continuation of factorized skew non-symmetric Macdonald coefficients.

  2. Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems

    hep-th 2026-01 unverdicted novelty 7.0 of 10

    For t = q^{-m}, eigenfunctions from DIM Hamiltonians and twisted Cherednik Hamiltonians combine into identical symmetric functions that are eigenfunctions of both systems simultaneously.

  3. Generating twisted Cherednik eigenfunctions

    hep-th 2026-02 conditional novelty 6.0 of 10

    Twisted Macdonald polynomials are generated recursively from a ground state by creation and permutation moves, proving three conjectures about their coefficients.

  4. Twisted Cherednik spectrum as a $q,t$-deformation

    hep-th 2026-01 unverdicted novelty 6.0 of 10

    The twisted Cherednik spectrum is a q,t-deformation of the polynomial eigenfunctions built from symmetric ground states and weak-composition excitations at q=1.

  5. A basic triad in Macdonald theory

    hep-th 2024-11 conditional novelty 6.0 of 10

    At t=q^{-m}, the Noumi-Shiraishi series reproduces the Baker-Akhiezer function, completing a triad with the Macdonald polynomials.

  6. Integrable systems inspired by DAHA and DIM algebra: type $C^\vee C$ versus type $A$

    hep-th 2026-07 accept novelty 4.5 of 10

    Type C∨C DAHA and Koornwinder systems mirror type-A Macdonald structures for Hamiltonians, recursions, evaluations and dualities, but lack a usable Noumi-Shiraishi-style universal series and SL(2,Z)-type twisting auto...

  7. Elliptic triad

    hep-th 2024-12 conditional novelty 4.0 of 10

    The paper argues that the Macdonald triad admits elliptic deformations, but the defining linear equations for elliptic Baker-Akhiezer functions remain ambiguous except at m=1.

  8. Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems

    hep-th 2024-11 conditional novelty 4.0 of 10

    In explicit small cases, twisted Baker-Akhiezer functions satisfy the defining linear equations and are eigenfunctions of the integer-ray DIM Hamiltonians.

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