Hall-Littlewood and Macdonald polynomials

These notes are when I guest lectured in Thomas Lam’s class on Symmetric Functions in Winter 2024. I modeled my notes off of Mark Haiman’s presentation of Hall-Littlewood polynomials and Macdonald polynomials for his Math 249 class in the Spring 2020 term at UC Berkeley. Contrasting from Macdonald’s book, in these talks I take as a starting point the Schur-positive Hall-Littlewood polynomials and define them in a way that is most easily compatible with the construction of “Catalanimals” from my research with collaborators. Many facts were not proven due to time constraints, but my hope is that these notes provide a nice overview about a good way to think about Hall-Littlewood and Macdonald polynomials.

References

  1. Blasiak, J., Haiman, M., Morse, J., Pun, A., & Seelinger, G. (2023). A raising operator formula for Macdonald polynomials. arXiv:2307.06517.
  2. Butler, L. M. (1994). Subgroup Lattices and Symmetric Functions. Memoirs of the American Mathematical Soc.
  3. Haglund, J., Haiman, M., & Loehr, N. (2005). A combinatorial formula for Macdonald polynomials. , 18(3), 735–761.
  4. Lascoux, A., & Schutzenberger, Marcel-Paul (1978). Sur une conjecture de HO Foulkes. CR Acad. Sci. Paris Ser. AB.
Date published: Thursday, March 21, 2024