Hivert polynomials

2026-07-03 Improve this page

#Hivert polynomials

Hivert polynomials were introduced by F. Hivert in [Hiv00], as a quasisymmetric analog of the Hall–Littlewood polynomials. These interpolate between the Gessel quasisymmetric functions and the monomial quasisymmetric functions.

Transition matrices involving Hivert polynomials are studied in [LSW13].

Let \(\alpha=(\alpha_1,\dotsc,\alpha_k)\) be a composition and let \(\beta\) be a refinement of \(\alpha.\) If the parts of \(\beta\) refining \(\alpha_j\) are counted by \(b_j,\) set \[\mathrm{Bre}(\beta,\alpha)=(b_1,\dotsc,b_k), \qquad s(\alpha,\beta)\coloneqq \sum_{j=1}^k j(b_j-1).\] One convenient normalization of the Hivert polynomial is \[\hivert_\alpha(\xvec;t) = \sum_{\beta\text{ refines }\alpha} (-1)^{\ell(\beta)-\ell(\alpha)} t^{s(\alpha,\beta)}\gessel_\beta(\xvec).\] This is [Thm. 11, LSW13], written as a definition. It gives \[\hivert_\alpha(\xvec;0)=\gessel_\alpha(\xvec), \qquad \hivert_\alpha(\xvec;1)=\qmonom_\alpha(\xvec).\]

Example

For \(\alpha=(3),\) the refinements are \((3),\) \((2,1),\) \((1,2),\) and \((1,1,1).\) Thus \[\hivert_3 = \gessel_3 -t\gessel_{21} -t\gessel_{12} +t^2\gessel_{111}.\] In the monomial quasisymmetric basis this becomes \[\hivert_3 = \qmonom_3 +(1-t)\qmonom_{21} +(1-t)\qmonom_{12} +(1-t)^2\qmonom_{111}.\] The inverse transition begins \[\gessel_3 = \hivert_3 +t\hivert_{21} +t\hivert_{12} +t^3\hivert_{111},\] which is the degree-three case of [Thm. 26, LSW13].

Hivert also introduced quasisymmetric divided-difference operators. Replacing the ordinary divided differences by Hivert’s operators in the construction of Schur functions recovers the fundamental quasisymmetric functions. A. Hicks and E. Niese [HN24] apply the same idea to key polynomials and Demazure atoms. The resulting analogues are the fundamental slide polynomials and the fundamental particle basis, respectively.

Bibliography

[HN24]Angela Hicks and Elizabeth Niese. Quasisymmetric divided difference operators and polynomial bases. arXiv:2406.02420, 2024.

.bib

@article{HicksNiese2024x,
  author = {Angela Hicks and Elizabeth Niese},
  title = {Quasisymmetric divided difference operators and polynomial bases},
  year = {2024},
  eprint = {2406.02420},
  url = {https://arxiv.org/abs/2406.02420},
  journal = {arXiv e-prints}
}

[Hiv00]Florent Hivert. Hecke algebras, difference operators, and quasi-symmetric functions. Advances in Mathematics, 155(2):181–238, November 2000.

.bib

@article{Hivert2000,
  doi = {10.1006/aima.1999.1901},
  url2 = {https://doi.org/10.1006/aima.1999.1901},
  year = {2000},
  month = nov,
  publisher = {Elsevier {BV}},
  volume = {155},
  number = {2},
  pages = {181--238},
  author = {Florent Hivert},
  title = {Hecke Algebras,  Difference Operators, and Quasi-Symmetric Functions},
  journal = {Advances in Mathematics}
}

[LSW13]Nicholas A. Loehr, Luis G. Serrano and Gregory S. Warrington. Transition matrices for symmetric and quasisymmetric Hall–Littlewood polynomials. Journal of Combinatorial Theory, Series A, 120(8):1996–2019, November 2013.

.bib

@article{LoehrSerranoWarrington2013,
  doi = {10.1016/j.jcta.2013.07.008},
  url2 = {https://doi.org/10.1016/j.jcta.2013.07.008},
  year = {2013},
  month = nov,
  publisher = {Elsevier {BV}},
  volume = {120},
  number = {8},
  pages = {1996--2019},
  author = {Nicholas A. Loehr and Luis G. Serrano and Gregory S. Warrington},
  title = {Transition matrices for symmetric and quasisymmetric {H}all--{L}ittlewood polynomials},
  journal = {Journal of Combinatorial Theory,  Series A}
}