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Peer-reviewedCC BYJournal fĆ¼r die reine und angewandte Mathematik (Crelles Journal)

On Macdonald expansions of š¯‘˛-chromatic symmetric functions and the Stanleyā€“Stembridge conjecture

  • Sean T. Griffin1
  • Anton Mellit2
  • Marino Romero3
  • Kevin Weigl2
  • Joshua Jeishing Wen2
  1. 1FakultƤt fĆ¼r Mathematik , UniversitƤt Wien , Wien , Austria . Current address: Department of Mathematics, University of North Texas, Denton, TX 76203, USA
  2. 2FakultƤt fĆ¼r Mathematik , UniversitƤt Wien , Wien , Austria
  3. 3Department of Mathematics and Statistics , Minnesota State University, Mankato , Mankato , MN 560001 , United States
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Abstract

The Stanleyā€“Stembridge conjecture asserts that the chromatic symmetric function of the incomparability graph of a ( 3 + 1 ) (3+1) -free poset is š¯‘’-positive. Recently, Hikita proved this conjecture by giving an explicit š¯‘’-expansion of the Shareshianā€“Wachs š¯‘˛-chromatic refinement for unit interval graphs. Using the A q , t \mathbb{A}_{q,t} algebra, we give an expansion of these š¯‘˛-chromatic symmetric functions into Macdonald polynomials. Upon setting t = 1 t=1 , we obtain another proof of the Stanleyā€“Stembridge conjecture and re-derive Hikitaā€™s formula. Upon setting t = 0 t=0 , we obtain an expansion into Hallā€“Littlewood symmetric functions.

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Cite

BibTeXRIS
BibTeX
@article{griffin2026macdonald,
  title = {{On Macdonald expansions of š¯‘˛-chromatic symmetric functions and the Stanleyā€“Stembridge conjecture}},
  author = {Sean T. Griffin and Anton Mellit and Marino Romero and Kevin Weigl and Joshua Jeishing Wen},
  journal = {Journal fĆ¼r die reine und angewandte Mathematik (Crelles Journal)},
  year = {2026},
  month = nov,
  doi = {10.1515/crelle-2026-0086},
  url = {https://doi.org/10.1515/crelle-2026-0086},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - On Macdonald expansions of š¯‘˛-chromatic symmetric functions and the Stanleyā€“Stembridge conjecture
AU  - Sean T. Griffin
AU  - Anton Mellit
AU  - Marino Romero
AU  - Kevin Weigl
AU  - Joshua Jeishing Wen
PY  - 2026
DA  - 2026/11/04
JO  - Journal fĆ¼r die reine und angewandte Mathematik (Crelles Journal)
DO  - 10.1515/crelle-2026-0086
UR  - https://doi.org/10.1515/crelle-2026-0086
AB  - The Stanleyā€“Stembridge conjecture asserts that the chromatic symmetric function of the incomparability graph of a ( 3 + 1 ) (3+1) -free poset is š¯‘’-positive. Recently, Hikita proved this conjecture by giving an explicit š¯‘’-expansion of the Shareshianā€“Wachs š¯‘˛-chromatic refinement for unit interval graphs. Using the A q , t \mathbb{A}_{q,t} algebra, we give an expansion of these š¯‘˛-chromatic symmetric functions into Macdonald polynomials. Upon setting t = 1 t=1 , we obtain another proof of the Stanleyā€“Stembridge conjecture and re-derive Hikitaā€™s formula. Upon setting t = 0 t=0 , we obtain an expansion into Hallā€“Littlewood symmetric functions.
SN  - 0075-4102
SN  - 1435-5345
ER  -