Noncommutative symmetric functions

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#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. They also describe the homology classes of torus-invariant subvarieties in toric Richardson varieties, where effective classes correspond exactly to ribbon-positive elements [GLL26, Thm. II].

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. González, and S. Márquez 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.

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