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Theta operators at t=1, Macdonald cumulants, and LLT positivity

  • Jim Haglund
  • Vasu Tewari
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Abstract

We study the Theta operators of D'Adderio-Iraci-Vanden Wyngaerd at t=1 and show that the power-sum-indexed operators Θpk|t=1 agree with a commuting family of derivations when restricted to symmetric functions of positive degree. Writing h̃a for the modified Macdonald function indexed by the single row (a) we establish that Θpμh̃a|t=1 is, up to a normalization, the single-row Macdonald cumulant of Dolęga. We use these results to show that Θpμh̃a|t=1 and Θeλh̃a|t=1 are both sums of vertical-strip LLT polynomials indexed by certain plane trees. The former further shows that single-row Macdonald cumulants are LLT-positive, thereby yielding a stronger form of the higher-order Macdonald positivity conjecture of Dolęga when all shapes are single rows.

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Cite

BibTeXRIS
BibTeX
@misc{haglund2026theta,
  title = {{Theta operators at $t=1$, Macdonald cumulants, and LLT positivity}},
  author = {Jim Haglund and Vasu Tewari},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29957},
  eprint = {2609.29957},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29957},
}
RIS
TY  - UNPB
TI  - Theta operators at t=1, Macdonald cumulants, and LLT positivity
AU  - Jim Haglund
AU  - Vasu Tewari
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29957
UR  - https://arxiv.org/abs/2609.29957
AB  - We study the Theta operators of D'Adderio-Iraci-Vanden Wyngaerd at t=1 and show that the power-sum-indexed operators Θ_p_k|_t=1 agree with a commuting family of derivations when restricted to symmetric functions of positive degree. Writing h̃_a for the modified Macdonald function indexed by the single row (a) we establish that Θ_p_μh̃_a|_t=1 is, up to a normalization, the single-row Macdonald cumulant of Dolęga. We use these results to show that Θ_p_μh̃_a|_t=1 and Θ_e_λh̃_a|_t=1 are both sums of vertical-strip LLT polynomials indexed by certain plane trees. The former further shows that single-row Macdonald cumulants are LLT-positive, thereby yielding a stronger form of the higher-order Macdonald positivity conjecture of Dolęga when all shapes are single rows.
ER  -