We introduce and study a generalization of Schur's $P$-/$Q$-functions associated to a polynomial sequence, which can be viewed as ``Macdonald's ninth variation'' for $P$-/$Q$-functions. This variation includes as special cases Schur's $P$-/$Q$-functions, Ivanov's factorial $P$-/$Q$-functions and the $t=-1$ specialization of Hall--Littlewood functions associated to the classical root systems. We establish several identities and properties such as generalizations of Schur's original definition of Schur's $Q$-functions, Cauchy-type identity, Józefiak--Pragacz--Nimmo formula for skew $Q$-functions, and Pieri-type rule for multiplication.
This paper is an excerpt from the author's 1968 PhD dissertation [Yale University, 1968] in which the (now) well-known result, commonly known as the Folkman-Rado-Sanders theorem, is proved. The proof uses (finite) alternating sums of integers and an 'iterated Ramsey theorem' in a way analogous to the proof of Schur's theorem using differences of integers and Ramsey's theorem for the coloring of the edges of a complete graph. The proof predates all others except J. Folkman, who based his proof on van der Waerden's theorem. The paper also contains the first published statement of the countable v…
We obtain a common generalization of two types of Sylvester formulas for compound determinants and its Pfaffian analogue. As applications, we give generalizations of the Giambelli identity to skew Schur functions and the Schur identity to Schur's skew $Q$-functions.
In the literature there are several determinant formulas for Schur functions: the Jacobi-Trudi formula, the dual Jacobi-Trudi formula, the Giambelli formula, the Lascoux-Pragacz formula, and the Hamel-Goulden formula, where the Hamel-Goulden formula implies the others. In this paper we use an identity proved by Bazin in 1851 to derive determinant identities involving Macdonald's 9th variation of Schur functions. As an application we prove a determinant identity for factorial Schur functions conjectured by Morales, Pak, and Panova. We also obtain a generalization of the Hamel-Goulden formula, w…
We introduce a quasisymmetric generalization of Berele and Regev's hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. We examine the combinatorics of the quasisymmetric hook Schur functions, providing a relationship to Gessel's fundamental quasisymmetric functions and an analogue of the Robinson-Schensted-Knuth algorithm. We also prove that the multiplication of quasisymmetric hook Schur functions with hook Schur functions behaves the same as the multiplication of quasisymmetric Schur functions with Schur funct…
A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's parametrization for the Schur polynomial. We generalize Tokuyama's formula to the Hall-Littlewood polynomials by extending Tokuyama's statistics. Our result, in addition to specializing to Tokuyama's result and the aforementioned classical results, also yields connections to the monomial symmetric function and a new deformation of Stanley's formula.
We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding…
We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic formulas for such bounds were known, and only for $a\in\mathbb{N}$. Using symbolic computation techniques, these results are extended here to arbitrary $a\in\mathbb{R}$. Furthermore, we give exact formulas for the minimum number of monochromatic Schur triples for $a=1,2,3,4$, and briefly discuss the case $0<a<1$.
Lam and Pylyavskyy introduced loop symmetric functions as a generalization of symmetric functions. They defined loop Schur functions as generating functions over semistandard tableaux with respect to a `colored weight,' and they proved a Jacobi--Trudi-style determinantal formula for these generating functions. We prove that loop Schur functions can be expressed as a ratio of `loop alternants,' extending the analogy with Schur functions. As an application, we give a new proof of the loop version of the Murnaghan--Nakayama rule.
Recently, Kim and Oh expressed the Selberg integral in terms of the number of Young books which are a generalization of standard Young tableaux of shifted staircase shape. In this paper the generating function for Young books according to major index statistic is considered. It is shown that this generating function can be written as a Jackson integral which gives a new $q$-Selberg integral. It is also shown that the new $q$-Selberg integral has an expression in terms of Schur functions.
Our results revolve around a new operation on partitions, which we call overlap. We prove two overlap identities for so-called Littlewood-Schur functions. Littlewood-Schur functions are a generalization of Schur functions, whose study was introduced by Littlewood. More concretely, the Littlewood-Schur function $LS_λ(X; Y)$ indexed by the partition $λ$ is a polynomial in the variables $X \cup Y$ that is symmetric in both $X$ and $Y$ separately. The first overlap identity represents $LS _λ(X; Y)$ as a sum over subsets of $X$, while the second overlap identity essentially represents $LS_λ(X; Y)$ …
Motivated by Sato and Mori's work on the Korteweg-de Vries (KdV) equation and the modified KdV equation, Mizukawa, Nakajima, and Yamada made a conjecture on 2-reduced Schur functions and Schur's Q-functions. The conjecture claims that certain sums of products of a Littlewood-Richardson coefficient and two 2-reduced Schur functions are equal to Schur's Q-functions up to a scalar multiple. In this paper we give a proof of the conjecture in cases which have not been proved yet. We introduce a new expression of Schur's Q-functions and use it to prove the conjecture. Combinatorics of the inverse Ko…
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 $\mathcal{L}_C$ that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions $NSym$ . This b…
Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes
利用Schützenberger的jeu de taquin组合地和利用Jacobi-Trudi行列式代数地建立了斜Schur函数差的一些新关系.这些关系导致了斜Schur函数的某些差是Schur正的结论。将这些结果应用于一个对称函数的基础上，涉及带状舒尔函数证实了舒尔正猜想由于麦克纳马拉的有效性。进一步的应用表明，Schubert类乘积的某些差是Schubert正的.
In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions. This was because they had constructed a partial such set that was not a basis. We answer their question by defining Schur functions in noncommuting variables using a noncommutative analogue of the Jacobi-Trudi determinant. Our Schur functions in NCSym map to classical Schur functions under commutation, and a subset of them indexed by set partitions forms a basis for NCSym. Amongst other prop…
We consider the problem of determining when the difference of two ribbon Schur functions is a single Schur function. We fully classify the five infinite families of pairs of ribbon Schur functions whose difference is a single Schur function with corresponding partition having at most two parts at least $2$. We also prove an identity for differences of ribbon Schur functions and we determine some necessary conditions for such a difference to be Schur-positive, depending on the distribution of $1$'s and the end row lengths.
Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial …
Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.
We give a combinatorial proof of a natural generalization of the Murnaghan-Nakayama rule to loop Schur functions. We also define shifted loop Schur functions and prove that they satisfy a similar relation.
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 classify the $Q$-multiplicity-free skew Schur $Q$-functions. Towards this result, we also provide new relations between the shifted Littlewood-Richardson coefficients.
In this work, we establish a general relationship between the enumeration of weighted directed paths and skew Schur functions, extending work by Bousquet-Mélou, who expressed generating functions of discrete excursions in terms of rectangular Schur functions.
We classify the $Q$-homogeneous skew Schur $Q$-functions, i.e., those of the form $Q_{λ/μ} = k \cdot Q_ν$. On the way we develop new tools that are useful also in the context of other classification problems for skew Schur $Q$-functions.
We give a combinatorial formula for LLT cumulants of unicellular unilevelled shapes in terms of parking functions and Cayley trees. Our formula implies previously conjectured Schur positivity.
Subsequently to the author's preceding paper, we give full proofs of some explicit formulas about factorizations of $K$-$k$-Schur functions associated with any multiple $k$-rectangles.
We investigate chromatic symmetric functions in the relation to the algebra $Γ$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $Γ$ in terms of chromatic symmetric functions. We also consider the $p$-positivity of skew Schur $Q$-functions and find a class of $p$-positive ribbon Schur $Q$-functions, making a conjecture that they are \emph{all}. We include many concrete computational results that support our conjecture.
We show that no tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function, thereby providing a counterexample to a conjecture from the paper "Schur and e-positivity of trees and cut vertices".
