Skew quasisymmetric Schur functions and 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 LC that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ:NSym→Sym. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ.As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.     

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, (q,p)-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.     

Multiplicity Free Expansions of Schur <Emphasis Type="Italic">P</Emphasis>-Functions

After deriving inequalities on coefficients arising in the expansion of a Schur P-function in terms of Schur functions we give criteria for when such expansions are multiplicity free. From here we study the multiplicity of an irreducible spin character of the twisted symmetric group in the product of a basic spin character with an irreducible character of the symmetric group, and determine when it is multiplicity free. Received February 28, 2005     

A k-tableau characterization of k-Schur functions

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.     

Differential operator specializations of noncommutative symmetric functions

Let K be any unital commutative Q-algebra and z=(z₁,…,zn) commutative or noncommutative free variables. Let t be a formal parameter which commutes with z and elements of K. We denote uniformly by K《z》 and K?t?《z》 the formal power series algebras of z over K and K?t?, respectively. For any α?1, let D[α]《z》 be the unital algebra generated by the differential operators of K《z》 which increase the degree in z by at least α−1 and the group of automorphisms Ft(z)=z−Ht(z) of K?t?《z》 with o(Ht(z))?α and Ht=0(z)=0. First, for any fixed α?1 and , we introduce five sequences of differential operators of K《z》 and show that their generating functions form an NCS (noncommutative symmetric) system [W. Zhao, Noncommutative symmetric systems over associative algebras, J. Pure Appl. Algebra 210 (2) (2007) 363-382] over the differential algebra D[α]《z》. Consequently, by the universal property of the NCS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348, MR1327096; see also hep-th/9407124], we obtain a family of Hopf algebra homomorphisms , which are also grading-preserving when Ft satisfies certain conditions. Note that the homomorphisms SFt above can also be viewed as specializations of NCSFs by the differential operators of K《z》. Secondly, we show that, in both commutative and noncommutative cases, this family SFt (with all n?1 and ) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions [I. Gessel, Multipartite P-partitions and inner products of skew Schur functions, in: Contemp. Math., vol. 34, 1984, pp. 289-301, MR0777705; C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (3) (1995) 967-982, MR1358493; Richard P. Stanley, Enumerative Combinatorics II, Cambridge University Press, 1999] are also discussed.     

Peak quasisymmetric functions and Eulerian enumeration

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. We diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions, and study the g-polynomial for Eulerian posets as an algebra homomorphism.     

Counting involutory, unimodal, and alternating signed permutations

In this work we count the number of involutory, unimodal, and alternating elements of the group of signed permutations B_n, and the group of even-signed permutations D_n. Recurrence relations, generating functions, and explicit formulas of the enumerating sequences are given.     

Equality of Schur and Skew Schur Functions

We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables. Received May 29, 2004     

Coincidences among skew Schur functions

New sufficient conditions and necessary conditions are developed for two skew diagrams to give rise to the same skew Schur function. The sufficient conditions come from a variety of new operations related to ribbons (also known as border strips or rim hooks). The necessary conditions relate to the extent of overlap among the rows or among the columns of the skew diagram.     

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.     

Maximal 0–1-fillings of moon polyominoes with restricted chain lengths and rc-graphs

We show that maximal 0–1-fillings of moon polynomials with restricted chain lengths can be identified with certain rc-graphs, also known as pipe dreams. In particular, this exhibits a connection between maximal 0–1-fillings of Ferrers shapes and Schubert polynomials. Moreover, it entails a bijective proof showing that the number of maximal fillings of a stack polyomino S with no north-east chains longer than k depends only on k and the multiset of column heights of S.     

Effective scalar products of D-finite symmetric functions

Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are D-finite. We extend Gessel's work by providing algorithms that compute differential equations, these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel's class. Examples of applications to k-regular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself.     

Real-variables characterization of generalized Stieltjes functions

We obtain a characterization of generalized Stieltjes functions of any order λ>0$\lambda >0$ in terms of inequalities for their derivatives on (0,∞)$(0,\infty )$. When λ=1$\lambda =1$, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.     

Cylindric skew Schur functions

Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining these motivations, we study cylindric skew Schur functions from the point of view of Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus giving an expansion of an arbitrary cylindric skew Schur function in terms of skew Schur functions. While we show that no non-trivial cylindric skew Schur functions are Schur-positive, we conjecture that this can be reconciled using the new concept of cylindric Schur-positivity.     

Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given.     

Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups

Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler?s exponential generating function formula for the Eulerian numbers (Shareshian and Wachs, 2010 [17]). They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics, which generalize the Shareshian–Wachs q-analog of Euler?s formula, formulas of Foata and Han, and a formula of Chow and Gessel.     

Codimensions of algebras and growth functions

Let A be an algebra over a field F of characteristic zero and let cn(A), , be its sequence of codimensions. We prove that if cn(A) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α>1, an F-algebra Aα such that exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.     

Testing for equality between two copulas

We develop a test of equality between two dependence structures estimated through empirical copulas. We provide inference for independent or paired samples. The multiplier central limit theorem is used for calculating p-values of the Cramér-von Mises test statistic. Finite sample properties are assessed with Monte Carlo experiments. We apply the testing procedure on empirical examples in finance, psychology, insurance and medicine.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.