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Peer-reviewedCC BY-NC-SA 4.0Symmetry, Integrability and Geometry : Methods and Applications

Vector valued Jack polynomials from scratch

  • Charles F. Dunkl
  • Jean-Gabriel Luque
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Abstract

Vector valued Jack polynomials associated to the symmetric group SN are polynomials with multiplicities in an irreducible module of SN and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading term. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N) and studied by one of the authors (C. Dunkl) in the specialization r=p=1 (\emph{ie.} for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C.Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc.. We give also a shifted version of the construction and we discuss about vanishing properties of the associated polynomials.

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Cite

BibTeXRIS
BibTeX
@article{dunkl2026vector,
  title = {{Vector valued Jack polynomials from scratch}},
  author = {Charles F. Dunkl and Jean-Gabriel Luque},
  journal = {Symmetry, Integrability and Geometry : Methods and Applications},
  year = {2026},
  month = oct,
  eprint = {1009.2366},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-00517258v1},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - JOUR
TI  - Vector valued Jack polynomials from scratch
AU  - Charles F. Dunkl
AU  - Jean-Gabriel Luque
PY  - 2026
DA  - 2026/10/05
JO  - Symmetry, Integrability and Geometry : Methods and Applications
UR  - https://hal.science/hal-00517258v1
AB  - Vector valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of S_N and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading term. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N) and studied by one of the authors (C. Dunkl) in the specialization r=p=1 (\emph{ie.} for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C.Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc.. We give also a shifted version of the construction and we discuss about vanishing properties of the associated polynomials.
SN  - 1815-0659
ER  -