Samuel K. Hsiao, T. Kyle Petersen
We show that with the appropriate choice of coproduct, the type B quasisymmetric functions form a Hopf algebra, and the recently introduced type B peak functions form a Hopf subalgebra.
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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 cofree. 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.
We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of 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.
In this paper, we construct the weak version of peak quasisymmetric functions inside the Hopf algebra of weak composition quasisymmetric functions (WCQSym) defined by Guo, Thibon and Yu. Weak peak quasisymmetric functions (WPQSym) are studied in several aspects. First we find a natural basis of WPQSym lifting Stembridge's peak functions. Then we confirm that WPQSym is a Hopf subalgebra of WCQSym by giving explicit multiplication, comultiplication and antipode formulas. By extending Stembridge's descent-to-peak maps, we also show that WPQSym is a Hopf quotient algebra of WCQSym. On the other ha…
Interpolation in Species and a Lift of the Hopf Algebra of r-Quasisymmetric Functions
本文通过插值方法将r-拟对称函数理论提升至霍普夫单子理论，探讨了自由单子与自由交换单子之间的插值方法。
We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric…
Multi-quasisymmetric functions with semigroup exponents, Hopf algebras and Rota-Baxter algebras
本文在Rota-Baxter代数框架下推广拟对称函数，定义多变元序列上的多拟对称函数空间MQSym及半群指数版本MQSym^E，构建其Hopf代数结构，并证明自由交换幺Rota-Baxter代数同构于MQSym^E的标量扩张。
We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid $H$, and an $H$-structure $h$ on a set $N$, there are proper colorings of $h$, generalizing graph colorings and poset partitions. We show that the automorphism group of $h$ acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley's chromatic symmetric function and specializes to the o…
Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-commuting Variables
本研究提出拟对称函数霍普夫代数的组合定义新P基（细化对称幂和基），将其推广至非交换变量情形并验证基元一致性，还探讨了基的推广方法。
We investigate the rigidity for the Hopf algebra ${\rm QSym}$ of quasisymmetric functions with respect to the monomial, the fundamental and the quasisymmetric Schur basis, respectively. By establishing some combinatorial properties of the posets of compositions arising from the analogous Pieri rules for quasisymmetric functions, we show that ${\rm QSym}$ is rigid as an algebra with respect to the quasisymmetric Schur basis, and rigid as a coalgebra with respect to the monomial and the quasisymmetric Schur basis, respectively. The natural actions of reversal, complement and transpose of the lab…
Hazewinkel proved the Ditters conjecture that the algebra of quasisymmetric functions over the integers is free commutative by constructing a nice polynomial basis. In this paper we prove a structure theorem for the algebra of peak quasisymmetric functions (PQSym) over the integers. It provides a polynomial basis of PQSym over the rational field, different from Hsiao's basis, and implies the freeness of PQSym over its subring of symmetric functions spanned by Schur's Q-functions.
This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This paper attempt to survey the ongoing investigations in this topic as dictated by the knowledge and interes…
For a generalized permutohedron $Q$ the enumerator $F(Q)$ of positive lattice points in interiors of maximal cones of the normal fan $Σ_Q$ is a quasisymmetric function. We describe this function for the class of nestohedra as a Hopf algebra morphism from a combinatorial Hopf algebra of building sets. For the class of graph-associahedra the corresponding quasisymmetric function is a new isomorphism invariant of graphs. The obtained invariant is quite natural as it is the generating function of ordered colorings of graphs and satisfies the recurrence relation with respect to deletions of vertice…
On the quasisymmetric functions in superspace
本文研究超空间中拟对称函数的代数结构，引入非交换变量下的拟对称函数代数，并构建超排列的Hopf超代数，推导了基本拟对称函数的乘积公式。
Shuffle Bases and Quasisymmetric Power Sums
本文通过无穷小特征刻画 shuffle 基，为 Shuffle 代数建立泛性质，并据此统一构造拟对称幂和，恢复文献中四种已知构造并研究其性质。
Expanding quasisymmetric Schur $Q$-functions into peak Young quasisymmetric Schur functions
本文研究 peak 代数中两类拟对称函数基的关系，证明拟对称 Schur $Q$-函数可按非负系数展开为 peak Young 拟对称 Schur 函数。
We introduce new bases for the Hopf algebra of quasisymmetric functions that refine the symmetric powersum basis. These bases are expanded in terms of quasisymmetric monomial functions by using fillings of matrices. We define the analog of these bases in quasisymmetric functions of non-commuting variables. Our new bases have a (shifted) shuffle product and a deconcatenate coproduct. Finally, we describe a change of basis rule from the quasisymmetric powersum basis to the quasisymmetric fundamental basis.
We introduce megagreedoids, which generalize polymatroids, megamatroids, and greedoids. We define a quasisymmetric function invariant for a megagreedoid, and show that it has a positive expansion in the basis of fundamental quasisymmetric functions. Our proof involves lexicographic shellability. We also show that megagreedoids form a Hopf monoid. A running example is a megagreedoid associated to a rooted connected graph, and the resulting generalization of the chromatic symmetric function.
We generalize the Hopf algebras of free quasisymmetric functions, quasisymmetric functions, noncommutative symmetric functions, and symmetric functions to certain representations of the category of all finite Coxeter systems and its dual category. We investigate their connections with the representation theory of 0-Hecke algebras of finite Coxeter systems. Restricted to type B and D we obtain dual graded modules and comodules over the corresponding Hopf algebras in type A.
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic…
A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant (1) defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, (2) is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid, (3) 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, (4) behaves simply under matroid duality, (5) ha…
On the Hopf algebra of noncommutative symmetric functions in superspace
本文研究了超空间中非交换对称函数的Hopf代数sNSym，通过引入本原元素、推广初等与幂和函数、构造非交换Ribbon Schur函数及其对偶基本拟对称函数，并证明sNSym可实现为树的Hopf代数。
This article serves as an introduction to several recent developments in the study of quasisymmetric functions. The focus of this survey is on connections between quasisymmetric functions and the combinatorial Hopf algebra of noncommutative symmetric functions, appearances of quasisymmetric functions within the theory of Macdonald polynomials, and analogues of symmetric functions. Topics include the significance of quasisymmetric functions in representation theory (such as representations of the 0-Hecke algebra), recently discovered bases (including analogues of well-studied symmetric function…
Via duality of Hopf algebras, there is a direct association between peak quasisymmetric functions and enumeration of chains in Eulerian posets. We study this association explicitly, showing that the notion of $\cd$-index, long studied in the context of convex polytopes and Eulerian posets, arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by Stembridge. Thus Eulerian posets having a nonnegative $\cd$-index (for example, face lattices of convex polytopes) correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis.…
Chromatic word-quasisymmetric functions of matroids
本文研究拟阵的色词拟对称函数，证明其可估值性，并利用特殊拟阵类给出映射秩的下界$2^d-d$，猜想上界$d!$紧。核心贡献是引入该映射并探讨其区分能力。
There is a very natural and well-behaved Hopf algebra morphism from quasisymmetric functions to peak algebra, which we call it Theta map. This paper focuses on generalizing the peak algebra by constructing generalized Theta maps for an arbitrary combinatorial Hopf algebra. The image of Theta maps lies in the odd Hopf subalgebras, so we present a strategy to find odd Hopf subalgebra of any combinatorial Hopf algebra. We also give a combinatorial description of a family of Theta maps for Malvenuto-Reutenauer Hopf algebra of permutations $\operatorname{\mathsf{\mathfrak{S}Sym}}$ whose images are …
In this note we derive skew Pieri rules in the spirit of Assaf-McNamara for skew quasisymmetric Schur functions using the Hopf algebraic techniques of Lam-Lauve-Sottile, and recover the original rules of Assaf-McNamara as a special case. We then apply these techniques a second time to obtain skew Pieri rules for skew noncommutative Schur functions.
Recently a new basis for the Hopf algebra of quasisymmetric functions $QSym$, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset $\mathcal{L}_C$ that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions $NSym$. This bas…
As a natural basis of the Hopf algebra of quasisymmetric functions, monomial quasisymmetric functions are formal power series defined from compositions. The same definition applies to left weak compositions, while leads to divergence for other weak compositions. We adapt the method of renormalization in quantum field theory, in the framework of Connes and Kreimer, to deal with such divergency. This approach defines monomial quasisymmetric functions for any weak composition as power series while extending the quasi-shuffle (stuffle) relation satisfied by the usual quasisymmetric functions. The …
For a hypergraphic polytope there is a weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines its f-polynomial. This quasisymmetric function invariant of hypergraphs extends the Stanley chromatic symmetric function of simple graphs. We consider a certain combinatorial Hopf algebra of hypergraphs and show that universal morphism to quasisymmetric functions coincides with this enumerator function.
Mason and Remmel introduced a basis for quasisymmetric functions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur functions. We introduce a modification known as Young row-strict quasisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric funct…
We give an in-depth study of the Tchebyshev transforms of the first and second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform (of the first kind) preserves desirable combinatorial properties, including Eulerianess (due to Hetyei) and EL-shellability. It is also a linear transformation on flag vectors. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg and Readdy omega map of oriented matroids. One consequence is that nonnegativity of the cd-index is maintained. The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on the …
Quasisymmetric Schur $Q$-functions and peak Young quasisymmetric Schur functions
本文探讨拟对称Schur $Q$-函数与峰值杨拟对称Schur函数的关系，通过引入双射结合算法证明过渡矩阵性质，给出钩长公式等组合性质，明确特定情况下的展开式。
We develop a theory of multigraded (i.e., $N^l$-graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar, Bergeron, and Sottile [Compos. Math. 142 (2006), 1--30]. In particular we introduce the notion of canonical $k$-odd and $k$-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our results are specific categorical results for higher level quasisymmetric functions, several basis change formulas, and a generalization of the descents-to-pea…
Pattern-Avoiding Peak Functions
本文针对依据峰函数定义的模式避免拟对称函数，精确刻画$\mathfrak{S}_3$中使其对称的子集，并给出Schur $Q$函数正展开的显式公式。
We generalize Stembridge's enriched $P$-partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group. Whereas Stembridge's enriched $P$-partitions are related to quasisymmetric functions (the coalgebra dual to Solomon's type A descent algebra), our generalized enriched $P$-partitions are related to type B quasisymmetric functions (the coalgebra dual to Solomon's type B descent algebra). Using these functions, we explore three different peak algebras: the "interior" and "left" peak algebras of type A, and a new type B peak algebr…
A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirr…
Generalized chromatic functions
本文为边划分有向图定义了顶点着色，由此提出广义色函数，统一了P-分划与图的色对称函数等理论，并构建了非交换变量中r-拟对称函数的新Hopf代数。
We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.
By Foissy's work, the bidendriform structure of the Word Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual. However, the only known explicit isomorphism due to Vargas does not respect the bidendriform structure. This structure is entirely determined by so-called totally primitive elements (elements such that the two half-coproducts vanish). In this paper, we construct two bases indexed by two new combinatorial families called red (dual side) and blue (primal side) biplane forests in bijection with packed words. In those bases, primitive elements are indexe…
Kromatic quasisymmetric functions
本文用组合 Hopf 代数构造图的 kromatic 对称函数，证明其正展开，并研究两个拟对称 $q$-模拟的对称性及正展开条件。
A Hopf algebra generalization of the symmetric functions in partially commutative variables
本文在部分交换变量中推广对称函数，构造Hopf代数$Sym_A$及其对偶$PSym_A$，定义经典基的推广并描述代数结构，最后给出推广Schur函数的对偶基及开放问题。
We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable. From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containin…
We introduce a general method for constructing modules for $0$-Hecke algebras and supermodules for $0$-Hecke-Clifford algebras from diagrams of boxes in the plane, and give formulas for the images of these modules in the algebras of quasisymmetric functions and peak functions under the relevant characteristic map. As initial applications, we resolve a question of Jing and Li (2015), introduce a new basis of the peak algebra analogous to the quasisymmetric Schur functions, uncover a new connection between Schur $Q$-functions and quasisymmetric Schur functions, give a representation-theoretic in…
Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generalizations such as the word Hopf algebra and the double word Hopf algebra (and many more).
We develop the theory of weighted P-partitions, which generalises the theory of P-partitions from labelled posets to weighted labelled posets. We define the related generating functions in the natural way and compute their product, coproduct and other properties. As an application we introduce the basis of combinatorial power sums for the Hopf algebra of quasisymmetric functions and the reverse basis, both of which refine the power sum symmetric functions. These bases share many properties with the type 1 and type 2 quasisymmetric power sums introduced by Ballantine, Daugherty, Hicks, Mason an…
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoretic analogues of the by now classical ``square'' of Hopf algebras consisting of symmetric functions, quasisymmetric functions, noncommutative symmetric functions and the Malvenuto-Reutenauer Hopf algebra of permutations. In addition, we develop a theory of set-valued P-partitions and study three new families of symmetric functions which are weight generating functions of reverse plane partitions, weak set-va…
We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasisymmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discuss…
The peak algebra in noncommuting variables
本文将下降到峰映射与峰代数推广到非交换变量，定义 NCQSym、NCΠ 及非交换 Schur Q-函数，并证明其经典类似性质与广义 Dehn-Sommerville 方程。
Poset modules of the $0$-Hecke algebras and related quasisymmetric power sum expansions
本文研究0-Hecke代数的偏序集模，证明相关Grothendieck群的直和具有拟对称函数Hopf代数结构，还分析Schur函数拟类似物的拟对称幂和展开，给出新算法并通过边界条表表示对偶无污函数等的展开系数。
Finite noncommutative coinvariant algebras
本文研究自由结合代数由对称、拟对称等多项式生成的双边理想的商代数，证明任意域上拟对称等共变式有限性，给出次数界，确定Hilbert级数等，特征零下维数增长快于n!的固定幂。
Quasisymmetric expansion of Hall-Littlewood symmetric functions
该研究基于此前提出的q-基本拟对称函数（插值于Gessel基本函数与Stembridge峰函数之间），证明其可给出参数t=-q的Hall-Littlewood S-对称函数的拟对称展开，丰富了对称函数的拟对称展开相关结果。
These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive self-adjoint Hopf algebra" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric function…
This paper constructs a novel Hopf algebra $\mathsf{cf}(\mathrm{UT}_{\bullet})$ on the class functions of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$ over a finite field. This construction is representation theoretic in nature and uses the machinery of Hopf monoids in the category of vector species. In contrast with a similar known construction, this Hopf algebra has the property that induction to the finite general linear group induces a homomorphism to Zelevinsky's Hopf algebra of $\mathrm{GL}_{n}(\mathbb{F}_{q})$ class functions. Furthermore, $\mathsf{cf}(\mathrm…
A generalization of the dual immaculate quasisymmetric functions in partially commutative variables
本文将无瑕基和对偶无瑕基推广至着色代数 $QSym_A$ 和 $NSym_A$，通过表组合和创建算子定义着色对偶无瑕与无瑕函数，研究其 Hopf 代数结构、基展开、右 Pieri 规则及斜函数，并扩展至行严格情形。
A quasisymmetric analog of Grassmannian Schubert varieties
本文证明格拉斯曼流形中环面理查森簇的上同调环为拟对称函数环的有限截断，构造仿射铺砌，并利用直接极限得到具有弱H-群结构的ind-簇，其上同调为拟对称函数的Hopf代数，同时推出碎片多面体的f-向量对数凹性。
Shifted combinatorial Hopf algebras from $K$-theory
该研究将此前与Lewis合作构建的峰拟对称函数Hopf代数的$K$-理论推广，纳入含六个Hopf代数的平移版$K$-理论组合Hopf代数图，得到经典峰代数的新$K$-理论类似物，还研究了相关$K$-理论Schur函数生成的Hopf代数并推导公式、提出开放问题。
We develop a more general view of Stembridge's enriched $P$-partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group. Initially we focus on commutative peak algebras, spanned by sums of permutations with the same number of peaks, where we consider several variations on the definition of "peak." Whereas Stembridge's enriched $P$-partitions are related to quasisymmetric functions (the dual coalgebra of Solomon's type A descent algebra), our generalized enriched $P$-partitions are related to type B quasisymmetric functions (the…
The algebra of extended peaks
本文在q-变形P-划分研究基础上，构造了拟对称函数环的一类新子代数，其基为q取单位复根时的Gessel基本拟对称函数q-类似物，并提出介于峰集与下降集之间的“扩展峰”统计量。
If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmertic functions in both the monomial and fundamental bases, the Hopf algebra of multi-quasisymmetric functions in the fundamental basis, and the incidenc…
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