Eigenvalues of Heckman–Polychronakos operators

Dunkl, Charles; Gorin, Vadim

doi:10.5802/alco.451

Dunkl, Charles ¹; Gorin, Vadim ²

¹ University of Virginia, Department of Mathematics, Charlottesville, VA 22904 (USA)
² University of California at Berkeley, Departments of Statistics and Mathematics, Berkeley, CA 94720 (USA)

Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1547-1566

Abstract

Heckman–Polychronakos operators form a prominent family of commuting differential-difference operators defined in terms of the Dunkl operators $\mathcal{D}_i$ as $\mathcal{P}_m= \sum _{i=1}^N (x_i \mathcal{D}_i)^m$. They have been known since 1990s in connection with trigonometric Calogero–Moser–Sutherland Hamiltonian and Jack symmetric polynomials. We explicitly compute the eigenvalues of these operators for symmetric and skew-symmetric eigenfunctions, as well as partial sums of eigenvalues for general polynomial eigenfunctions.

Received: 2025-03-26
Revised: 2025-08-11
Accepted: 2025-08-14
Published online: 2026-01-06

DOI: 10.5802/alco.451

Classification: 05E05, 05E10, 20C30, 15A18, 81Q05
Keywords: Jack polynomial, operators on polynomials, characters of representations, symmetric group

Author's affiliations:

Dunkl, Charles ¹; Gorin, Vadim ²

¹ University of Virginia, Department of Mathematics, Charlottesville, VA 22904 (USA)
² University of California at Berkeley, Departments of Statistics and Mathematics, Berkeley, CA 94720 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Dunkl, Charles; Gorin, Vadim. Eigenvalues of Heckman–Polychronakos operators. Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1547-1566. doi: 10.5802/alco.451

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