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PreprintCC BY 4.0arXiv

On the Nazarov-Sklyanin Lax operator for Macdonald functions

  • Ryan Mickler
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Abstract

We study the Lax operator for Macdonald symmetric functions introduced by Nazarov and Sklyanin in 2015, extending the spectral theory developed by Mickler-Moll in the Jack case. We extend the NS operator to a Lax operator acting on symmetric functions with coefficients in polynomials in an auxiliary variable, and prove it has an orthogonal eigenbasis indexed by partitions and their addable corners, with the corner contents as eigenvalues, and leading terms given by Macdonald symmetric functions. We obtain the Macdonald analogue of the author's earlier Jack box-sum identity for Littlewood-Richardson coefficients. We then strengthen this box-sum identity to a two-parameter formula, proved using a finite-rank product formula of Warnaar. Its residues give explicit linear relations among Macdonald Littlewood-Richardson coefficients.

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Cite

BibTeXRIS
BibTeX
@misc{mickler2026nazarovsklyanin,
  title = {{On the Nazarov-Sklyanin Lax operator for Macdonald functions}},
  author = {Ryan Mickler},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12326},
  eprint = {2610.12326},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12326},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Nazarov-Sklyanin Lax operator for Macdonald functions
AU  - Ryan Mickler
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12326
UR  - https://arxiv.org/abs/2610.12326
AB  - We study the Lax operator for Macdonald symmetric functions introduced by Nazarov and Sklyanin in 2015, extending the spectral theory developed by Mickler-Moll in the Jack case. We extend the NS operator to a Lax operator acting on symmetric functions with coefficients in polynomials in an auxiliary variable, and prove it has an orthogonal eigenbasis indexed by partitions and their addable corners, with the corner contents as eigenvalues, and leading terms given by Macdonald symmetric functions. We obtain the Macdonald analogue of the author's earlier Jack box-sum identity for Littlewood-Richardson coefficients. We then strengthen this box-sum identity to a two-parameter formula, proved using a finite-rank product formula of Warnaar. Its residues give explicit linear relations among Macdonald Littlewood-Richardson coefficients.
ER  -