We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian Gr_{SL_k} into Schubert homology classes in Gr_{SL_{k+1}}. This is achieved by studying the combinatorics of a new class of partitions called k-shapes, which interpolates between k-cores and k+1-cores. We define a symmetric function for each k-shape, and show that they expand positively in terms of dual k-Schur functions. We obtain an explicit combinatorial description of the expansion of an ungraded k-Schur function into k+1-Schur functions. As a corollary, we give a formula for the Schur expansion of an ungraded k-Schur function.
In 1995 Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$, whose $e$-positivity and Schur-positivity has been of large interest. In this paper we study the relative $e$-positivity and Schur-positivity between connected graphs on $n$ vertices. We define and investigate two families of posets on distinct chromatic symmetric functions. The relations depend on the $e$-positivity or Schur-positivity of a weighed subtraction between $X_G$ and $X_H$. We find a biconditional condition between $e$-positivity or Schur-positivity and the relation to the complete graph. This gives a…
The `Weyl symmetric functions' studied here naturally generalize classical symmetric (polynomial) functions, and `Weyl bialternants,' sometimes also called Weyl characters, analogize the Schur functions. For this generalization, the underlying symmetry group is a finite Weyl group. A `splitting poset' for a Weyl bialternant is an edge-colored ranked poset possessing a certain structural property and a natural weighting of its elements so that the weighted sum of poset elements is the given Weyl bialternant. Connected such posets are of combinatorial interest in part because they are rank symme…
We consider families of quasisymmetric functions with the property that if a symmetric function $f$ is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of $f$. We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families. This poset includes the Schur functions, the quasisymmetric Schur functions, the fundamental quasisym…
The machinery of noncommutative Schur functions is a general approach to Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang recently adapted this theory to posets to give a new approach to the Stanley-Stembridge conjecture. We further develop this theory to prove that the symmetric function associated to any $P$-Knuth equivalence graph is Schur positive. This settles a conjecture of Kim and the third author, and refines results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of chromatic symmetric functions.
We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions, and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters and Demazure atoms; the quasi-key, fundamental and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule…
Young's lattice is a prototypical example of differential posets. Differential posets have the Robinson correspondence, the correspondence between permutations and pairs of standard tableaux with the same shape, as in the case of Young's lattice. Fomin introduced generalized Schur operators to generalize the method of Robinson correspondence in differential posets to the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard tableaux with the same shape. In this paper, we introduce operators on the vector space whose basis is the set of …
In prior work, the authors, along with M. McClard, R. A. Proctor, and N. J. Wildberger, studied certain distributive lattice models for the `Weyl bialternants' (aka `Weyl characters') associated with the rank two root systems/Weyl groups. These distributive lattices were uniformly described as lattices of order ideals taken from certain grid-like posets, although the arguments connecting the lattices to Weyl bialternants were case-by-case depending on the type of the rank two root system. Using this connection with Weyl bialternants, these lattices were shown to be rank symmetric and rank unim…
By establishing relations between operators on compositions, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded graphs and dual filtered graphs when paired with the poset of compositions arising from the Pieri rules for quasisymmetric Schur functions and its deformation.
A poset on a certain class of partitions known as k-shapes was recently introduced to provide a combinatorial rule for the expansion of a (k-1)-Schur functions into k-Schur functions at t=1. The main ingredient in this construction was a bijection, which we call the weak bijection, that associates to a k-tableau a pair made out of a (k-1)-tableau and a path in the poset of k-shapes. We define here a concept of charge on k-tableaux (which conjecturally gives a combinatorial interpretation for the expansion coefficients of Hall-Littlewood polynomials into k-Schur functions), and show that it is …
For a $(3+1)$-free poset $P$, we define a hybrid of $P$-tableaux and cylindric tableaux called cylindric $P$-tableaux. We introduce $P$-analogs of cylindric Schur functions, defined by a determinantal formula, and prove that they are the weight generating functions of cylindric $P$-tableaux. We deduce that certain sums of the $e$-expansion coefficients of the chromatic symmetric function $X_{inc(P)}$ are positive. This improves on Gasharov's theorem on the Schur positivity of $X_{inc(P)}$ and gives further evidence for the Stanley-Stembridge conjecture.
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…
We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a polynomial expression for their drift. We also recover Rietsch's parametriza-tion of totally nonnegative unitriangular Toeplitz matrices without using quantum cohomology of flag varieties. All the homeomorphisms we define can moreover be made explicit by using the combinatorics of k-Schur functions and elementary computations based on Perron-Frobenius theorem.
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…
The branching of (k-1)-Schur functions into k-Schur functions was given by Lapointe, Lam, Morse and Shimozono as chains in a poset on k-shapes. The k-Schur functions are the parameterless case of a more general family of symmetric functions over Q(t), conjectured to satisfy a k-branching formula given by weights on the k-shape poset. A concept of a (co)charge on a k-tableau was defined by Lapointe and Pinto. Although it is not manifestly positive, they prove it is compatible with the k-shape poset for standard k-tableau and the positivity follows. Morse introduced a manifestly positive notion …
Upper homogeneous finite type (upho) posets are a large class of partially ordered sets with the property that the principal order filter at every vertex is isomorphic to the whole poset. Well-known examples include k-array trees, the grid graphs, and the Stern poset. Very little is known about upho posets in general. In this paper, we construct upho posets with Schur-positive Ehrenborg quasisymmetric functions, whose rank-generating functions have rational poles and zeros. We also categorize the rank-generating functions of all planar upho posets. Finally, we prove the existence of an upho po…
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…
Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight λ, we define a preorder on the set $P(λ, k)$ of k-tuples of dominant weights which add up to λ. Let $P(λ, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if λis a multiple of a fundamental weight (and k is general) or if k=2 (and λis general), then $P(λ, k)/\sim$ coincides with the set of S_k-orbits in $P(λ,k)$, where S_k acts on $P(λ, k)$ as the permutations of components. If g is of type A_n and k=2, we show that the S_2-orbit of the row …
The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual $k$-Schur funct…
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 give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{μ\leλ} g^{(k)}_μ$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by $\widetilde{g}^{(k)}_λ$. As an application of this, we also give a $k$-rectangle factorization formula $\widetilde{g}^{(k)}_{R_t\cupλ}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_λ$ where $R_t=(t^{k+1-t})$, analogous to that of $k$-Schur functions $s^{(k)}_{R_t\cupλ}=s^{(k)}_{R_t}s^{(k)}_λ$.
To every labeled poset (P,ω), one can associate a quasisymmetric generating function for its (P,ω)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew Schur function equality is still open, a complete classification of equality among (P,ω) generating functions is likely too much to expect. Instead, we determine necessary conditions and separate sufficient conditions for two labeled posets to have equal generating functions. We conclude with a classification of all equalities for labeled posets with small numbers of line…
We give combinatorial proofs that certain families of differences of products of Schur functions are monomial-positive. We show in addition that such monomial-positivity is to be expected of a large class of generating functions with combinatorial definitions similar to Schur functions. These generating functions are defined on posets with labelled Hasse diagrams and include for example generating functions of Stanley's $(P,ω)$-partitions.
The chromatic symmetric function of a graph is a generalization of the chromatic polynomial. The key motivation for studying the structure of a chromatic symmetric function is to answer positivity conjectures by Stanley in 1995 and Gasharov in 1996. As a symmetric function one can write the chromatic symmetric function in the basis of Schur functions. In this paper we address the positivity of the Schur coefficients when the parameter of the Schur function is a hook shape. Furthermore, when a graph is the incomparability graph of a poset with specific properties we construct a correspondence b…
We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of $(\mathbb{C}^d)^{\otimes n}$. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a maj…
We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur functions and resolve the Schur positivity and $k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.
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…
We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Toeplitz matrices. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for $E$-polynomials of series-parallel posets and column-strict labelled Ferrers posets.
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…
This article is dedicated to the study of positivity phenomena for the chromatic symmetric function of a graph with respect to various bases of symmetric functions. We give a new proof of Gasharov's theorem on the Schur-positivity of the chromatic symmetric function of a $(3 + 1)$-free poset. We present a combinatorial interpretation of the Schur-coefficients in terms of planar networks. Compared to Gasharov's proof, it gives a clearer visual illustration of the cancellation procedures and is quite similar in spirit to the proof of monomial positivity of Schur functions via the Lindstrom-Gesse…
We prove that the $K$-$k$-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. Lam-Schilling-Shimozono identified the $K$-$k$-Schur functions as Schubert representatives for $K$-homology of the affine Grassmannian for SL$_{k+1}$. Our perspective reveals that the $K$-$k$-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a …
We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of $\mathfrak{sl}(n,\mathbb{C})$ obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for each finite-dimensional representation of $\mathfrak{sl}(n,\mathbb{C})$ associated to a given skew Schur function. Our constructions use diamond-colored distributive lattices of skew-shaped semistandard tableaux that generalize some classical Gelfand--Tsetlin (GT) lattices. Our constructions take place within the context of a certain programmatic study of p…
We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the $q$-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimp…
University of Pennsylvania（宾夕法尼亚大学）；University of California, Berkeley（加州大学伯克利分校）；CNRS, IMJ-PRG, Sorbonne Université（法国国家科学研究中心，IMJ-PRG，索邦大学）；Reed College（里德学院）
We show the equivalence of the Pieri formula for flag manifolds and certain identities among the structure constants, giving new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric function, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtaining a new…
We review some algebraic and combinatorial structures that underlie models in the KPZ universality class.Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these combinatorial constructions are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also present how fundamental representation theoretic concepts, such a…
Two methods of constructing 2D Toda $τ$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group $S_n$ by multiplication of the conjugacy class sums $C_λ\in C[S_n]$ in the group algebra by elements of an abelian group of central elements. Extending the characteristic map to the tensor product $C[S_n]\otimes C[S_n]$ leads to double expansions in terms of power sum symmetric functions, in which the coefficients count the number of such paths. Applying the sam…
