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 flag h-vector of Eulerian posets.
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.
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. …
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 2007, McNamara proved that two skew shapes can have the same Schur support only if they have the same number of $k\times \ell$ rectangles as subdiagrams. This implies that two ribbons can have the same Schur support only if one is obtained by permuting row lengths of the other. We present substantial progress towards classifying when a permutation $π\in S_m$ of row lengths of a ribbon $α$ produces a ribbon $α_π$ with the same Schur support as $α$; when this occurs for all $π\in S_m$, we say that $α$ has "full equivalence class." Our main results include a sufficient condition for a ribbon $…
McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that $B\leq _s A$ if $s_A-s_B$ is Schur-positive. Towards this, McNamara and van Willigenburg proved that it suffices to study equitable ribbons, namely ribbons whose row lengths are all of length $a$ or $(a+1)$ for $a\geq 2$. In this paper we confirm the conjecture of McNamara and Pylyavskyy in all cases where the comparable equitable ribbons form a chain. We also confirm a conjecture of McNamara and van Willigenburg regarding which equitable ribbons in general are m…
We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specializes to (extended) $k$-deletion, and we give two methods to obtain numerous new bases from weighted graphs for the algebra of symmetric functions. Moreover, we classify when two weighted paths have equal extended chromatic symmetric functions by proving this is equivalent to the classification of equal ribbon Schur functions. This latter classification is dependent on the operation composition of compositions, which we generalize to composition of graphs. We then appl…
We consider ribbon shapes, not necessarily connected, whose rows, with at least two boxes in each, are in monotone length order. These ribbons are uniquely defined by a pair of partitions: the row partition consisting of the row lengths in decreasing order, and the overlapping partition whose entries count the total number of columns with two boxes in the successive ribbon shapes obtained by sequentially subtracting the longest row. The support of such ribbon Schur functions, considered as a subposet of the dominance order lattice on partitions, has the row partition as bottom element, and, as…
Towards equivalent thickened ribbon Schur functions
School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）；Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics（甘肃省数学与统计学基础学科研究中心）
本文受 Morse--Seelinger 猜想启发，为 C 型强标记表格建立置换模型，通过 C 型换位与对称核模型建立保持标记的显式对应，并描述仿射 Weyl 群在有界划分上的作用。
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.
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…
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 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…
In 1995, the first author introduced a multivariate generating function {$G$} that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that $G$ is symmetric, he conjectured that $G$ is Schur positive. We prove this conjecture by expanding $G$ positively in terms of ribbon Schur functions. We obtain this expansion using a weight-preserving bijection whose inverse is inspired by the Push-Glide algorithm of Préville-Ratelle and Viennot. In fact, this weight-preserving bijection allows us to establish a stronger version of the first author's conjecture s…
We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms $s_ν[h_λ]$ and $s_ν[e_λ]$, for any partitions $λ$ and $ν$. We first give general results about this approach, then use results on tableaux, ribbon tableaux and integer points in polytopes to understand the case where $ν$ is a partition of $3$ and $λ$ has one part. We then use a \textit{Kronecker map} to extend these results to any partition $λ$.
The purpose of this note is to introduce a new family of quasi-symmetric functions called LLT cumulants and discuss its properties. We define LLT cumulants using the algebraic framework for conditional cumulants and we prove that the Macdonald cumulant has an explicit positive expansion in terms of LLT cumulants of ribbon shapes, generalizing the classical decomposition of Macdonald polynomials. We also find a natural combinatorial interpretation of the LLT cumulant of a given directed graph as a weighted generating function of colorings of its subgraphs. We use this graph theoretical framewor…
A combinatorial expression for the coefficient of the Schur function $s_λ$ in the expansion of the plethysm $p_{n/d}^d \circ s_μ$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $λ$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_μ, s_λ \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_μ^n, s_λ \rangle$-element set under the $d^{\text{th}}$ power of an order-$n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $λ$ is rectangular, …
