Quasisymmetric Functions Research Articles

A computational and combinatorial exposé of plethystic calculus

In recent years, plethystic calculus has emerged as a powerful technical tool for studying symmetric polynomials. In particular, some striking recent advances in the theory of Macdonald polynomials have relied heavily on plethystic computations. The main purpose of this article is to give a detailed explanation of a method for finding combinatorial interpretations of many commonly occurring plethystic expressions, which utilizes expansions in terms of quasisymmetric functions. To aid newcomers to plethysm, we also provide a self-contained exposition of the fundamental computational rules underlying plethystic calculus. Although these rules are well-known, their proofs can be difficult to extract from the literature. Our treatment emphasizes concrete calculations and the central role played by evaluation homomorphisms arising from the universal mapping property for polynomial rings.

From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix

Every symmetric function f can be written uniquely as a linear combination of Schur functions, say f = ∑ λ x λ s λ , and also as a linear combination of fundamental quasisymmetric functions, say f = ∑ α y α Q α . For many choices of f arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients y α and wishes to compute the Schur coefficients x λ . This paper gives a general combinatorial formula expressing each x λ as a linear combination of the y α ’s, where each coefficient in this linear combination is + 1 , − 1 , or 0. This formula arises by suitably modifying Eğecioğlu and Remmel’s combinatorial interpretation of the inverse Kostka matrix involving special rim-hook tableaux.

Quadrilaterals, extremal quasiconformal extensions and Hamilton sequences

The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations K0(h) and K1(h) of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for K0(h) = K1(h) when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.

The Tchebyshev Transforms of the First and Second Kind

An in-depth study of the Tchebyshev transforms of the first and second kind of a poset is taken. The Tchebyshev transform of the first kind is shown to preserve desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg, and Readdy omega map of oriented matroids. The Tchebyshev transform of the second kind U is a Hopf algebra endomorphism on the space of quasisymmetric functions which, when restricted to Eulerian posets, coincides with Stembridge’s peak enumerator. The complete spectrum of U is determined, generalizing the work of Billera, Hsiao, and van Willigenburg. The type B quasisymmetric function of a poset is introduced and, like Ehrenborg’s classical quasisymmetric function of a poset, it is a comodule morphism with respect to the quasisymmetric functions QSym. Finally, similarities among the omega map, Ehrenborg’s r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps which occur naturally in the setting of combinatorial Hopf algebras.

Colored Posets and Colored Quasisymmetric Functions

The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach, we introduce colored analogs of P-partitions and enriched P-partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile’s theory of combinatorial Hopf algebras and its colored analog.

Valuative invariants for polymatroids

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera–Jia–Reiner quasi-symmetric function, and the invariant G introduced by the first author, are valuative. In this paper we construct the Z -modules of all Z -valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the Z -modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that G is universal for valuative invariants.

Colored Trees and Noncommutative Symmetric Functions

Let ${\cal CRF}_S$ denote the category of $S$-colored rooted forests, and H$_{{\cal CRF}_S}$ denote its Ringel-Hall algebra as introduced by Kremnizer and Szczesny. We construct a homomorphism from a $K^+_0({\cal CRF}_S)$–graded version of the Hopf algebra of noncommutative symmetric functions to H$_{{\cal CRF}_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0({\cal CRF}_S)$–graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao.

A three-parameter Hopf deformation of the algebra of Feynman-like diagrams*

We construct a three-parameter deformation of the Hopf algebra LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the product formula in a simplified version of quantum field theory. This new algebra is a true Hopf deformation which reduces to LDIAG for some parameter values and to the algebra of matrix quasi-symmetric functions (MQSym) for others, and thus relates LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler–Zagier sums.

Valuative invariants for polymatroids

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal{G}$ introduced by the first author, are valuative. In this paper we construct the $\mathbb{Z}$-modules of all $\mathbb{Z}$-valued valuative functions for labelled matroids and polymatroids on a fixed ground set, and their unlabelled counterparts, the $\mathbb{Z}$-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that $\mathcal{G}$ is universal for valuative invariants. Beaucoup des invariants importants des matroïdes et polymatroïdes, tels que le polynôme de Tutte, la fonction quasi-symmetrique de Billera-Jia-Reiner, et l'invariant $\mathcal{G}$ introduit par le premier auteur, sont valuatifs. Dans cet article nous construisons les $\mathbb{Z}$-modules de fonctions valuatives aux valeurs entières des matroïdes et polymatroïdes étiquetés définis sur un ensemble fixe, et leurs équivalents pas étiquetés, les $\mathbb{Z}$-modules des invariants valuatifs. Nous fournissons des bases des ces modules et leurs modules duels, engendrés par fonctions caractéristiques des polytopes, et des formules explicites donnant leurs rangs. Nos résultats confirment une conjecture du premier auteur, que $\mathcal{G}$ soit universel pour les invariants valuatifs.

Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall–Littlewood theory.

Constructing quasisymmetric functions via graph-directed iterated function systems

Using Tukia’s method for representing a quasisymmetric function as a quasisymmetric sieve, we generalize his modification to the Salem scheme and find a sufficient condition for the collection of functions that realize a structure parametrization of a graph-directed function system of a particular form (a one-dimensional multizipper) to consist of quasisymmetric functions. We give an asymptotically sharp estimate for the quasisymmetry coefficient of these functions in terms of the dilation coefficients of the mappings constituting a given multizipper, which yields a substantially more general method for constructing quasisymmetric functions than Tukia’s construction.

THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and co-free. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.

Inversion of some series of free quasi-symmetric functions

We give a combinatorial formula for the inverses of the alternating sums of free quasi-symmetric functions of the form F ω ( I ) where I runs over compositions with parts in a prescribed set C . This proves in particular three special cases (no restriction, even parts, and all parts equal to 2) which were conjectured by B.C.V. Ung in [B.C.V. Ung, Combinatorial identities for series of quasi-symmetric functions, in: Proc. FPSAC’08, Toronto, 2008].

Quasi-symmetric functions and the KP hierarchy

Quasi-symmetric functions arise in an approach to solve the Kadomtsev–Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product and the outer coproduct. In particular, supplied with this new product and the outer coproduct, the algebra of quasi-symmetric functions becomes an infinitesimal bialgebra. Using these results we derive a sequence of identities in the algebra of quasi-symmetric functions that are in formal correspondence with the equations of the KP hierarchy.

Random walks on quasisymmetric functions

Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained. Several important random walks are now realized this way: Stanley's QS-distribution results from endomorphisms given by evaluation maps, a-shuffles result from the ath convolution power of the universal character, and the Tchebyshev operator of the second kind introduced recently by Ehrenborg and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg regarding the spectra for a family of random walks on ab-words is proven. A theorem of Stembridge from the theory of enriched P-partitions is also recovered as a special case.

Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions

We give a new representation theoretic interpretation of the ring of quasi-symmetric functions. This is obtained by showing that the super analogue of Gessel's fundamental quasi-symmetric function can be realized as the character of a connected crystal for the Lie superalgebra gl n | n associated to its non-standard Borel subalgebra with a maximal number of odd isotropic simple roots. We also present an algebraic characterization of these super quasi-symmetric functions.

Générateurs Explicites pour les Fonctions Quasisymétriques sur les Entiers Rationels

In (Hazewinkel in Adv. Math. 164:283–300, 2001, and CWI preprint, 2001) it has been proved that the ring of quasisymmetric functions over the integers is free polynomial. This is a matter that has been of great interest since 1972; for instance because of the role this statement plays in a classification theory for noncommutative formal groups that has been in development since then, see (Ditters in Invent. Math. 17:1–20, 1972; in Scholtens’ Thesis, Free Univ. of Amsterdam, 1996) and the references in the latter. Meanwhile quasisymmetric functions have found many more applications (see Gel’fand et al. in Adv. Math. 112:218–348, 1995). However, the proofs of the author in the aforementioned papers do not give explicit polynomial generators for QSymm over the integers. In this note I give a (really quite simple) set of polynomial generators for QSymm over the integers.

A quasisymmetric function for matroids

A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant: • defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients; • is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid; • is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight; • behaves simply under matroid duality; • has a simple expansion in terms of P -partition enumerators; • is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising from the work of Lafforgue, where lack of such a decomposition implies that the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis.

A one-parameter family of dendriform identities

We prove a q-identity in the dendriform algebra of colored free quasi-symmetric functions. For q = 1 , we recover identities due to Ebrahimi-Fard, Manchon, and Patras, in particular the noncommutative Bohnenblust–Spitzer identity.

Quasisymmetric functions from a topological point of view

It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions $\mathsf {Symm}$. We offer the cohomology of the space $\Omega \Sigma {\mathsf C} P^{\infty}$ as a topological model for the ring of quasisymmetric functions $\mathsf {QSymm}$. We exploit standard results from topology to shed light on some of the algebraic properties of $\mathsf {QSymm}$. In particular, we reprove the Ditters conjecture. We investigate a product on $\Omega \Sigma {\mathsf C} P^{\infty}$ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of $BU$. The canonical Thom spectrum over $\Omega \Sigma {\mathsf C} P^{\infty}$ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.