Noncommutative symmetric functions

2026-08-17 Improve this page

#Noncommutative symmetric functions

For an introduction to noncommutative symmetric functions, we refer to [GKLL+95].

The algebra of noncommutative symmetric functions is dual to the algebra of quasisymmetric functions. These are not to be confused with symmetric functions in noncommuting variables. One convenient model is \[\mathrm{NSym}=\setZ\langle \completeH_1,\completeH_2,\dotsc\rangle,\] the free associative algebra generated by the noncommutative complete homogeneous functions. For a composition $\alpha,$ write $\completeH_\alpha\coloneqq \completeH_{\alpha_1}\completeH_{\alpha_2}\dotsm \completeH_{\alpha_\ell}.$ The pairing with quasisymmetric functions is normalized by $\langle \completeH_\alpha,\qmonom_\beta\rangle=\delta_{\alpha,\beta}.$ There is a forgetful morphism $\chi:\mathrm{NSym}\to\spaceSym$ sending $\completeH_n$ to the ordinary complete homogeneous symmetric function.

There are therefore two complementary maps to keep in mind: \[\mathrm{NSym} \xrightarrow{\ \chi\ } \spaceSym, \qquad \mathrm{NSym} \cong \spaceQSym^\ast .\] The first map forgets noncommutativity. The second identifies NSym with the graded dual of $\spaceQSym,$ so that every change of basis in QSym has a transpose change of basis in NSym.

E. Chen, K. Ono, and M. Mogielnicki lift the sprout symmetric function for record compositions of alternating permutations to NSym using noncommutative power sums [COM26]. Their expansion gives exact formulas for the number of alternating permutations with a prescribed record composition.

#Dual bases and the forgetful map

The basic dual pairs are: $ \text{NSym basis} $ $ \text{Dual QSym basis} $ $ \text{Image under }\chi$ $ \completeH_\alpha $ $ \qmonom_\alpha $ $ \completeH_{\alpha_1}\completeH_{\alpha_2}\dotsm \completeH_{\alpha_\ell}$ $ R_\alpha $ $ \gessel_\alpha $ $ \text{ordinary ribbon Schur function}$ $ \mathfrak S_\alpha $ $ \text{dual immaculate} $ $ \schurS_\alpha\text{ when }\alpha\text{ is a partition}$ $ \Psi_\alpha,\Phi_\alpha $ $ \qPsi_\alpha,\qPhi_\alpha $ $ \text{ordinary power-sum expressions}$ Here $R_\alpha$ is the noncommutative ribbon basis. The ribbon basis is dual to the fundamental quasisymmetric basis, so it is the NSym side of descent-set enumerators. This is why ribbons appear naturally in the 0-Hecke categorification of QSym and NSym.

The two noncommutative power-sum bases above are dual to the two quasisymmetric power-sum bases $\qPsi$ and $\qPhi.$ Since $\qPsi_\alpha$ and $\qPhi_\alpha$ have positive expansions in the monomial quasisymmetric basis, duality gives positive expansions of $\completeH_\alpha$ in the corresponding noncommutative power-sum bases [BDHM+20].

A. Hicks and R. McCloskey give a concrete realization of $\mathrm{NSym}$ in noncommuting variables and derive the standard noncommutative elementary, complete, ribbon, and power-sum bases from that model [HM24]. Their account also gives combinatorial change-of-basis formulas using minimal products of transition matrices and brick tabloid statistics, paralleling the classical symmetric function picture.

J. M. Campbell introduces a lift of Stanley’s chromatic symmetric function to $\mathrm{NSym}$ [Cam24]. For an unlabeled directed graph $D,$ the construction gives an element $X_D\in \mathrm{NSym}$ whose commutative image is the ordinary chromatic symmetric function of the underlying undirected graph. The lift is obtained from Stanley’s power-sum expansion using the $\Psi$-basis of $\mathrm{NSym},$ and it leads to digraph-indexed generating sets for $\mathrm{NSym}.$ Thus the orientation data is retained at the noncommutative level and then forgotten by the projection $\chi:\mathrm{NSym}\to\spaceSym.$ L. Hao and S. Zhu show that Campbell’s chromatic noncommutative symmetric function distinguishes oriented stars, oriented double stars, and several families of oriented caterpillars and paths [HZ26]. Thus retaining the orientation before applying $\chi$ can recover graph data lost by the ordinary chromatic symmetric function. J.-C. Novelli and J.-Y. Thibon study chromatic quasisymmetric functions inside the algebra of quasisymmetric functions in noncommuting variables [NT25]. Their work also proposes noncommutative Macdonald polynomials compatible with a Haglund–Wilson type formula and relates the construction to Yang–Baxter elements in Hecke algebras.

E. E. Allen and S. K. Mason give a combinatorial formula for the expansion of immaculate noncommutative symmetric functions in the complete homogeneous noncommutative basis [AM25]. Their model uses GBPR diagrams and tunnel hooks, extending the role played by special rim hooks in the classical inverse Kostka formula.

D. Arcis, C. Gonz{\'a}lez, and S. M{\'a}rquez study the Hopf algebra of noncommutative symmetric functions in superspace [AGM24]. They construct primitive elements, extend the elementary and power-sum families to superspace, introduce noncommutative ribbon Schur functions in superspace, and realize the Hopf algebra using trees. F. Lehner, J.-C. Novelli, and J.-Y. Thibon study combinatorial Hopf algebras in noncommutative probability [LNT20]. L. Guo, J.-Y. Thibon, and H. Yu study Hopf algebras of signed permutations, weak quasisymmetric functions, and Malvenuto–Reutenauer type structures [GTY20]. S. Zemel gives explicit antipode formulas for commutative and noncommutative $q$-deformations of QSym, including partial noncommutative-QSym and word-quasisymmetric variants [Zem26]. The usual QSym antipode formula is recovered by specialization.

#Noncommutative Schur functions

There are several Schur-like bases in $\mathrm{NSym}.$ The noncommutative Schur functions of C. Bessenrodt, K. Luoto, and S. v. Willigenburg are the basis dual to the quasisymmetric Schur functions [BLW11]. Equivalently, they give a composition-indexed lift of ordinary Schur functions under the forgetful map $\chi:\mathrm{NSym}\to\spaceSym.$ The older ribbon basis of [GKLL+95] is another fundamental Schur-like basis of $\mathrm{NSym}.$

#Immaculate Schur functions

The immaculate Schur functions form a composition-indexed basis of $\mathrm{NSym}$ introduced by C. Berg, N. Bergeron, F. Saliola, L. Serrano, and M. Zabrocki [BBSS+14]. They are constructed using Bernstein-like creation operators, and the forgetful map $\chi:\mathrm{NSym}\to\spaceSym$ sends $\mathfrak{S}_\lambda$ to the ordinary Schur function $\schurS_\lambda$ when $\lambda$ is a partition.

The immaculate basis has a positive right Pieri rule and a Jacobi–Trudi type formula. Its dual basis is the dual immaculate Schur functions in QSym.

#Murnaghan–Nakayama rule

V. Tewari proves a noncommutative analogue of the classical Murnaghan–Nakayama rule for the noncommutative Schur functions of C. Bessenrodt, K. Luoto, and S. v. Willigenburg [Tew16]. The rule expands a product of a noncommutative power sum and a noncommutative Schur function into noncommutative Schur functions, with signs controlled by noncommutative border strips.

Bibliography

[AM25]Edward E. Allen and Sarah Katherine Mason. A combinatorial interpretation of the noncommutative inverse kostka matrix. The Electronic Journal of Combinatorics, 32(1), 2025.

.bib

@article{AllenMason2025,
  author = {Edward E. Allen and Sarah Katherine Mason},
  title = {A combinatorial interpretation of the noncommutative inverse
    Kostka matrix},
  year = {2025},
  journal = {The Electronic Journal of Combinatorics},
  volume = {32},
  number = {1},
  doi = {10.37236/12677},
  url = {https://doi.org/10.37236/12677},
  eprint = {2207.05903}
}

[AGM24]Diego Arcis, Camilo González and Sebastián Márquez. On the hopf algebra of noncommutative symmetric functions in superspace. The Electronic Journal of Combinatorics, 31(3), 2024.

.bib

@article{ArcisGonzalezMarquez2024,
  author = {Diego Arcis and Camilo Gonz{\'a}lez and Sebasti{\'a}n M{\'a}rquez},
  title = {On the Hopf algebra of noncommutative symmetric functions in superspace},
  year = {2024},
  journal = {The Electronic Journal of Combinatorics},
  volume = {31},
  number = {3},
  doi = {10.37236/12295},
  url = {https://doi.org/10.37236/12295},
  eprint = {2205.11813}
}

[BDHM+20]Cristina Ballantine, Zajj Daugherty, Angela Hicks, Sarah Mason and Elizabeth Niese. On quasisymmetric power sums. Journal of Combinatorial Theory, Series A, 175:105273, October 2020.

.bib

@article{BallantineDaughertyHicksMason2020,
  doi = {10.1016/j.jcta.2020.105273},
  url2 = {https://doi.org/10.1016/j.jcta.2020.105273},
  year = {2020},
  month = oct,
  publisher = {Elsevier {BV}},
  volume = {175},
  pages = {105273},
  author = {Cristina Ballantine and Zajj Daugherty and Angela Hicks and Sarah Mason and Elizabeth Niese},
  title = {On quasisymmetric power sums},
  journal = {Journal of Combinatorial Theory,  Series A}
}

[BBSS+14]Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano and Mike Zabrocki. A lift of the Schur and Hall–Littlewood bases to non-commutative symmetric functions. Canadian Journal of Mathematics, 66(3):525–565, June 2014.

.bib

@article{BergBergeronSaliolaSerranoZabrocki2014,
  doi = {10.4153/cjm-2013-013-0},
  url2 = {https://doi.org/10.4153/cjm-2013-013-0},
  year = {2014},
  month = jun,
  publisher = {Canadian Mathematical Society},
  volume = {66},
  number = {3},
  pages = {525--565},
  author = {Chris Berg and Nantel Bergeron and Franco Saliola and Luis Serrano and Mike Zabrocki},
  title = {A Lift of the {S}chur and {H}all--{L}ittlewood Bases to Non-commutative Symmetric Functions},
  journal = {Canadian Journal of Mathematics}
}

[BLW11]C. Bessenrodt, K. Luoto and S. Willigenburg. Skew quasisymmetric Schur functions and noncommutative Schur functions. Advances in Mathematics, 226(5):4492–4532, March 2011.

.bib

@article{BessenrodtLuotoWilligenburg2011,
  doi = {10.1016/j.aim.2010.12.015},
  url = {https://doi.org/10.1016/j.aim.2010.12.015},
  year = {2011},
  month = mar,
  publisher = {Elsevier {BV}},
  volume = {226},
  number = {5},
  pages = {4492--4532},
  author = {C. Bessenrodt and K. Luoto and S. van Willigenburg},
  title = {Skew quasisymmetric {S}chur functions and noncommutative {S}chur functions},
  journal = {Advances in Mathematics}
}

[Cam24]John M. Campbell. A lift of chromatic symmetric functions to NSym. arXiv:2410.04669, 2024.

.bib

@article{Campbell2024x,
  author = {John M. Campbell},
  title = {A lift of chromatic symmetric functions to {NSym}},
  year = {2024},
  eprint = {2410.04669},
  url = {https://arxiv.org/abs/2410.04669},
  journal = {arXiv e-prints}
}

[COM26]Evan Chen, Ken Ono and Michal Mogielnicki. Record compositions of alternating permutations and noncommutative symmetric functions. arXiv:2607.12873v1, 2026.

.bib

@article{ChenOnoMogielnicki2026x,
  author = {Evan Chen and Ken Ono and Michal Mogielnicki},
  title = {Record compositions of alternating permutations and noncommutative symmetric functions},
  year = {2026},
  eprint = {2607.12873v1},
  url = {https://arxiv.org/abs/2607.12873v1},
  journal = {arXiv e-prints}
}

[GKLL+95]Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh and Jean-Yves Thibon. Noncommutative symmetrical functions. Advances in Mathematics, 112(2):218–348, May 1995.

.bib

@article{GelfandKrobLascouxLeclercRetakhThibon1995,
  doi = {10.1006/aima.1995.1032},
  url2 = {https://doi.org/10.1006/aima.1995.1032},
  year  = {1995},
  month = may,
  publisher = {Elsevier {BV}},
  volume = {112},
  number = {2},
  pages = {218--348},
  author = {Israel M. Gelfand and Daniel Krob and Alain Lascoux and Bernard Leclerc and Vladimir S. Retakh and Jean-Yves Thibon},
  title = {Noncommutative Symmetrical Functions},
  journal = {Advances in Mathematics}
}

[GTY20]Li Guo, Jean-Yves Thibon and Houyi Yu. The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto–Reutenauer. Advances in Mathematics, 374:107341, 2020.

.bib

@article{GuoThibonYu2020,
  author = {Li Guo and Jean-Yves Thibon and Houyi Yu},
  title = {The {H}opf algebras of signed permutations, of weak
    quasi-symmetric functions and of {M}alvenuto--{R}eutenauer},
  year = {2020},
  journal = {Advances in Mathematics},
  volume = {374},
  pages = {107341},
  doi = {10.1016/j.aim.2020.107341},
  eprint = {1912.12721}
}

[HZ26]Lingxiao Hao and Shenglin Zhu. The chromatic noncommutative symmetric function of oriented trees. arXiv:2606.03436v2, 2026.

.bib

@article{HaoZhu2026x,
  author = {Lingxiao Hao and Shenglin Zhu},
  title = {The chromatic noncommutative symmetric function of oriented trees},
  year = {2026},
  eprint = {2606.03436v2},
  url = {https://arxiv.org/abs/2606.03436v2},
  journal = {arXiv e-prints}
}

[HM24]Angela Hicks and Robert McCloskey. A Combinatorial Perspective on the Noncommutative Symmetric Functions. arXiv:2406.13728, 2024.

.bib

@article{HicksMcCloskey2024x,
  author = {Angela Hicks and Robert McCloskey},
  title = {A {C}ombinatorial {P}erspective on the {N}oncommutative
    {S}ymmetric {F}unctions},
  year = {2024},
  eprint = {2406.13728},
  url = {https://arxiv.org/abs/2406.13728},
  journal = {arXiv e-prints}
}

[LNT20]Franz Lehner, Jean-Christophe Novelli and Jean-Yves Thibon. Combinatorial Hopf algebras in noncommutative probability. arXiv:2006.02089, 2020.

.bib

@article{LehnerNovelliThibon2020x,
  author = {Franz Lehner and Jean-Christophe Novelli and Jean-Yves Thibon},
  title = {Combinatorial {H}opf algebras in noncommutative probability},
  year = {2020},
  eprint = {2006.02089},
  archivePrefix = {arXiv},
  primaryClass = {math.CO}
}

[NT25]Jean-Christophe Novelli and Jean-Yves Thibon. Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation. arXiv:2502.09072, 2025.

.bib

@article{NovelliThibon2025x,
  author = {Jean-Christophe Novelli and Jean-Yves Thibon},
  title = {Noncommutative chromatic quasi-symmetric functions, {M}acdonald polynomials, and the {Y}ang-{B}axter equation},
  year = {2025},
  eprint = {2502.09072},
  url = {https://arxiv.org/abs/2502.09072},
  journal = {arXiv e-prints}
}

[Tew16]Vasu Tewari. A Murnaghan–Nakayama rule for noncommutative Schur functions. European Journal of Combinatorics, 58:118–143, November 2016.

.bib

@article{Tewari2016,
  doi = {10.1016/j.ejc.2016.05.010},
  url2 = {https://doi.org/10.1016/j.ejc.2016.05.010},
  year  = {2016},
  month = nov,
  publisher = {Elsevier {BV}},
  volume = {58},
  pages = {118--143},
  author = {Vasu Tewari},
  title = {A {M}urnaghan--{N}akayama rule for noncommutative {S}chur functions},
  journal = {European Journal of Combinatorics},
  eprint = {1403.0607}
}

[Zem26]Shaul Zemel. The Antipodes of $q$-Quasi-Symmetric Functions and Non-Commutative Quasi-Symmetric Functions. arXiv:2607.07870v1, 2026.

.bib

@article{Zemel2026x,
  author = {Shaul Zemel},
  title = {The {A}ntipodes of $q$-{Q}uasi-{S}ymmetric {F}unctions and
    {N}on-{C}ommutative {Q}uasi-{S}ymmetric {F}unctions},
  year = {2026},
  eprint = {2607.07870v1},
  url = {https://arxiv.org/abs/2607.07870v1},
  journal = {arXiv e-prints}
}