Victor Reiner, Kristin M. Shaw, Stephanie van Willigenburg
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.
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We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables.
We classify the $Q$-multiplicity-free skew Schur $Q$-functions. Towards this result, we also provide new relations between the shifted Littlewood-Richardson coefficients.
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 g…
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 show that sequences of skew Schur polynomials obtained from stretched semi-standard Young tableaux satisfy a linear recurrence, which we give explicitly. Using this, we apply this to finding certain asymptotic behavior of these Schur polynomials and present conjectures on minimal recurrences for stretched Schur polynomials.
In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape…
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.
Some new relations on skew Schur function differences are established both combinatorially using Schützenberger's jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validity of a Schur positivity conjecture due to McNamara. A further application reveals that certain differences of products of Schubert classes are Schubert positive.
We present a single operation for constructing skew diagrams whose corresponding skew Schur functions are equal. This combinatorial operation naturally generalises and unifies all results of this type to date. Moreover, our operation suggests a closely related condition that we conjecture is necessary and sufficient for skew diagrams to yield equal skew Schur functions.
We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with López Martín and Teixidor i Bigas concerning Brill-Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner-Tenner-Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms o…
Skew Hives, Skew Skeps, Skew Schur Log-Concavity
该研究引入斜蜂巢与斜筐模型，证明推广Lam等人猜想的斜舒尔对数凹性结果，获Newell-Littlewood数等的对数凹性结论，并解答Speyer关于相关组合结构双射的问题。
We find a simple criterion for the equality $Q_λ=Q_{μ/ν}$ where $Q_λ$ and $Q_{μ/ν}$ are Schur's Q-functions on infinitely many variables.
Normalized skew Schur polynomials are Lorentzian
该研究证明有限变量下斜对称舒尔多项式的标准化形式为洛伦兹型，通过将其实现为舒伯特多项式特化并结合对偶雅可比-崔奇恒等式，推导出斜对称科斯塔数满足沿根方向的对数凹性不等式。
The hive model is a combinatorial device that may be used to determine Littlewood-Richardson coefficients and study their properties. It represents an alternative to the use of the Littlewood-Richardson rule. Here properties of hives are used to determine all possible multiplicity-free Schur function products and skew Schur function expansions. This confirms the results of Stembridge, Gutschwager, and Thomas and Yong, and sheds light on the combinatorial origin of the conditions for being multiplicity-free, as well as illustrating some of the key features and power of the hive model.
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 introduce a new operation on skew diagrams called composition of transpositions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur Q-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur Q-functions. Moreover, we determine all relations between ribbon Schur Q-functions; show they supply a Z-basis for skew Schur Q-functions; assert their irreducibility; and show that the non-commutative analogue of ribbon Schur Q-functions is the…
We consider the skew diagram $Δ_n$, which is the $180^\circ$ rotation of the staircase diagram $δ_n = (n,n-1,n-2,...,2,1)$. We create a staircase with bad foundation by augmenting $Δ_n$ with another skew diagram, which we call the \textit{foundation}. We consider pairs of staircases with bad foundation whose foundations are transposes of one another. Among these pairs, we show that the difference of the corresponding skew Schur functions is Schur-positive in the case when one of the foundations consists of either a one or two row diagram, or a hook diagram.
A concise proof of cylindric Schur positivity
本文通过简短的证明展示了斜圆柱Schur函数可以正向展开为非斜圆柱Schur函数，且展开系数为融合系数。
While equality of skew Schur functions is well understood, the problem of determining when two skew Schur $Q$ functions are equal is still largely open. It has been studied in the case of ribbon shapes in 2008 by Barekat and van Willigenburg, and this paper approaches the problem for near-ribbon shapes, formed by adding one box to a ribbon skew shape. We particularly consider frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. We conjecture, with evidence, that all Schur $Q$ functions of frayed ribbon shape are distinct up to antipodal reflection. …
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…
Richardson volume models for skew Schur and skew Schur $P/Q$-functions
该研究将斜Schur函数、斜Schur P/Q函数对应到Richardson簇的体积多项式，解决相关洛伦兹猜想并推广结果，还推导了多种不等式与多面体相关结论，补充了特征2下的态射结果。
Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalizing those of Josefiak-Pragacz, Nimmo, and most recently Okada. As a preamble to this we present a parallel develo…
The Schur-positivity order on skew shapes is defined by B \leq A if the difference s_A - s_B is Schur-positive. It is an open problem to determine those connected skew shapes that are maximal with respect to this ordering. A strong necessary condition for the Schur-positivity of s_A - s_B is that the support of B is contained in that of A, where the support of B is defined to be the set of partitions lambda for which s_lambda appears in the Schur expansion of s_B. We show that to determine the maximal connected skew shapes in the Schur-positivity order and this support containment order, it su…
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 define a fat staircase to be a Ferrers diagram corresponding to a partition of the form $(n^{α_n}, {n-1}^{α_{n-1}},..., 1^{α_1})$, where $α= (α_1,...,α_n)$ is a composition, or the $180^\circ$ rotation of such a diagram. If a diagram's skew Schur function is a linear combination of Schur functions of fat staircases, we call the diagram a sum of fat staircases. We prove a Schur-positivity result that is obtained each time we augment a sum of fat staircases with a skew diagram. We also determine conditions on which diagrams can be sums of fat staircases, including necessary and sufficient con…
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.
The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood-Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van…
The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case.
We define a fat staircase to be a Ferrers diagram corresponding to a partition of the form $(n^{α_n}, {n-1}^{α_{n-1}},..., 1^{α_1})$, where $α= (α_1,...,α_n)$ is a composition, or the $180^\circ$ rotation of such a diagram. We look at collections of skew diagrams consisting of a fixed fat staircase augmented with all hooks of a given size. Among these diagrams we determine precisely which pairs give a Schur-positive difference. We extend this classification to collections of fat staircases augmented with hook-complements.
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.
The determinantal identities of Hamel and Goulden have recently been shown to apply to a tableau-based ninth variation of skew Schur functions. Here we extend this approach and its results to the analogous tableau-based ninth variation of supersymmetric skew Schur functions. These tableaux are built on entries taken from an alphabet of unprimed and primed numbers and that may be ordered in a myriad of different ways, each leading to a determinantal identity. At the level of the ninth variation the corresponding determinantal identities are all distinct but the original notion of supersymmetry …
Equality of skew Schur functions in noncommuting variables
本文解决了非交换变量中两个斜 Schur 函数何时相等的分类问题，给出了当斜图不同时二者相等当且仅当一个为非对称 ribbon、另一个为其对径旋转且置换满足显式双射条件的完整刻画。
The question of when two skew Young diagrams produce the same skew Schur function has been well-studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the K-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons.
We provide a non-recursive, combinatorial classification of multiplicity-free skew Schur polynomials. These polynomials are $GL_n$, and $SL_n$, characters of the skew Schur modules. Our result extends work of H. Thomas--A. Yong, and C. Gutschwager, in which they classify the multiplicity-free skew Schur functions.
We prove Stanley's conjecture that, if delta_n is the staircase shape, then the skew Schur functions s_{delta_n / mu} are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function s_{delta_n / delta_{n-2}}, we discuss connections with Eulerian numbers and alternating permutations.
Reiner, Shaw and van Willigenburg showed that if two skew Schur functions s_A and s_B are equal, then the skew shapes A and B must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than skew Schur equality: that s_A and s_B have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of s_A contains that of s_B, then the row o…
In this paper we describe the thickened strips and the outside nested decompositions of any skew shape $λ/μ$. For any such decomposition $Φ=(Θ_1,Θ_2,\ldots,Θ_g)$ of the skew shape $λ/μ$ where $Θ_i$ is a thickened strip for every $i$, if $r$ is the number of boxes that are contained in any two distinct thickened strips of $Φ$, we establish a determinantal formula of the function $s_{λ/μ}(X)p_{1^r}(X)$ with the Schur functions of thickened strips as entries, where $s_{λ/μ}(X)$ is the Schur function of the skew shape $λ/μ$ and $p_{1^r}(X)$ is the power sum symmetric function index by the partitio…
Special values of $K$-theoretic Schur $P$- and $Q$-functions
本文给出斜版本 $K$-理论 Schur $P$/$Q$ 函数的特殊值，据此证明移位集值斜表数量的奇偶性并推广至其他斜情形，进而构造表之间的配对。
We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Outside decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determinantal expressions for the skew Schur functions including the Jacobi-Trudi determinant, its dual, the Giambelli determinant and the rim ribbon determinant due to Lascoux and Pragacz. Using cutting strips, one obtains a formula for the number of outside…
The diagonal derivative of a skew Schur polynomial
本文给出斜舒尔多项式在对角微分算子作用下的显式公式，推广了 Weigandt 对普通舒尔多项式的结果。
Cylindric skew Schur functions, a generalization of skew Schur functions, are closely related to the famous problem finding a combinatorial formula for the 3-point Gromov-Witten invariants of Grassmannian. In this paper, we prove cylindric Schur positivity of the cylindric skew Schur functions, conjectured by McNamara. We also show that all coefficients appearing in the expansion are the same as $3$-point Gromov-Witten invariants. We start discussing properties of affine Stanley symmetric functions for general affine permutations and $321$-avoiding affine permutations, and explain how these fu…
It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew shape. We characterise skew Schur functions (and therefore the product of two Schur functions) which are multiplicity-free and the resulting Schur expansion runs over the whole interval of partitions, i.e. skew Schur functions having Littlewood-Richardson coefficients always equal to $1$ over the full interval.
Flagged Skew Schur Polynomials Twisted By Roots Of Unity
本研究推广了Littlewood的Schur多项式因式分解定理，证明一类带标志斜Schur多项式在变量被单位根扭转时也有类似因式分解，且包含Demazure特征标作为特例。
We define a number of related combinatorial objects, each of which possesses a surprising symmetry. We include several applications such as a combinatorial explanation for certain fixed points of the involution $ω$ on the ring of symmetric functions, as well as a relationship between certain skew Schur functions and skew $Q$-Schur functions. We give a $t$-deformation of these $Q$-Schur functions, and show that it is Schur positive, including a combinatorial description of the Schur coefficients. A corollary of our results is the equality of skew $Q$-Schur functions: $Q_{λ+δ/μ+ δ}=Q_{λ'+δ/μ' + …
Symmetric quasisymmetric Schur-like functions
本文研究了行严格双重完美函数和扩展Schur函数及其斜泛化在何种情况下对称，以及其自然变体先进函数的对称性，结果恢复了经典斜Schur函数。
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.
In this paper, we extend recent results of Assaf and McNamara on skew Pieri rule and skew Murnaghan-Nakayama rule to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf-McNamara's original proof, and one via Lam-Lauve-Sotille's skew Littlewood-Richardson rule. We end with some conjectures for skew rules for Hall-Littlewood polynomials.
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.
Let $k$ and $m$ be positive integers and $λ/μ$ a skew partition. We compute the principal specialization of the skew Schur polynomials $s_{λ/μ}(x_1, \ldots, x_{k})$ modulo $q^m-1$ under suitable conditions. We interpret the results thus obtained from the viewpoint of the cyclic sieving phenomenon on semistandard Young skew tableaux of shape $λ/μ$. As an application, we deal with evaluations of the principal specialization of the skew Schur polynomials at roots of unity.
We show that the shifted rank, or srank, of any partition $λ$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_λ$ function in terms of power sum symmetric functions. This gives an affirmative answer to a conjecture of Clifford. As pointed out by Clifford, the notion of the srank can be naturally extended to a skew partition $λ/μ$ as the minimum number of bars among the corresponding skew bar tableaux. While the srank conjecture is not valid for skew partitions, we give an algorithm to compute the srank.
By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli type matrices of a skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This gives an affirmative answer to an open problem posed by Kuperberg.
Young's lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young's lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new pos…
In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference s_A - s_B of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Our conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for s_A = s_B, and we deduce a strengthening of their result as a special case.
We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.
Towards equivalent thickened ribbon Schur functions
本文研究连通斜分图的等价类，刻画了含单个2×m或m×2方格块的此类等价类大小，证实了McNamara等人2009年提出的相关猜想的特殊情况。
We use Hopf algebras to prove a version of the Littlewood-Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood-Richardson rules for Schur P- and Q-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.
There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that…
A non-iterative straightening algorithm and orthogonality for skew Schur modules
本文将 Fulton 的 Schur 模行列式构造推广至斜设置，提出了一种非迭代拉直算法以显式闭系数公式将任意元素表示为新的 D-基，并证明该基源于对半标准表基的 Gram-Schmidt 正交化，从而揭示了自然内积并将拉直重表述为正交投影。
Skew Symplectic and Orthogonal Schur Functions
本文利用辛与正交 Schur 函数的顶点算子表示定义并刻画斜辛、斜正交 Schur 函数，证明其分支规则并导出 Jacobi–Trudi、Gelfand–Tsetlin 与 Cauchy 型结果，实现 A、C、D 型的统一构造。
Pfaffian--Toeplitz identities, Schur positivity, and the $q$-log-convexity of Baxter polynomials
该研究利用Pfaffian与Toeplitz相关公式，证明Baxter多项式序列具q-对数凸性，其q-精细Baxter数阵列行与列均为q-对数凹，相关结果可推广至d-Hoggatt数的q-模拟。
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