A new combinatorial approach to the ribbon tableaux generating functions and q-Littlewood Richardson coefficients of Lascoux, Leclerc and Thibon is suggested. We define operators which add ribbons to partitions and following Fomin and Greene study non-commutative symmetric functions in these operators. This allows us to give combinatorial interpretations for some (skew) q-Littlewood Richardson coefficients whose non-negativity appears not to be known. Our set up also leads to a new proof of the action of the Heisenberg algebra on the Fock space of U_q(sl^_n) due to Kashiwara, Miwa and Stern.
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.
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…
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…
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. …
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.
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 define an equivalence relation on integer compositions and show that two ribbon Schur functions are identical if and only if their defining compositions are equivalent in this sense. This equivalence is completely determined by means of a factorization for compositions: equivalent compositions have factorizations that differ only by reversing some of the terms. As an application, we can derive identities on certain Littlewood-Richardson coefficients. Finally, we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis. We…
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）
Inspired by the spin-inversion statistics of Schilling et al. and Haglund et al., we relate the symmetry of ribbon functions to a result of van Leeuwen, and also describe the multiplication of a domino function by a Schur function.
We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomial…
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…
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.
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.
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 study a representation of the (local) plactic monoid given by Schur operators $u_i$, which act on partitions by adding a box in column $i$ (if possible). In particular, we give a complete list of the relations that hold in the algebra of Schur operators.
Lascoux, Leclerc and Thibon have introduced symmetric functions which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these `ribbon functions' in analogy with Schur functions. In particular we will describe ribbon Pieri and Murnagham-Nakayama formulae, a ribbon Cauchy identity and an algebra involution which `conjugates' the ribbon functions. We will study these functions in the context of the action of the Heisenberg algebra on the Fock space representation of the quantum affine algebra U_q(sl_n)^, discovered by Kashiwara, Miwa and Stern. We wil…
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 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.
This paper deals with decreasing operators on back stable Schubert polynomials. We study two operators $ξ$ and $\nabla$ of degree $-1$, which satisfy the Leibniz rule. Furthermore, we show that all other such operators are linear combinations of $ξ$ and $\nabla$. For the case of Schur functions, these two operators fully determine the product of Schur functions, i.e., it is possible to define the Littlewood-Richardson coefficients only from $ξ$ and $\nabla$. This new point of view on Schur functions gives us an elementary proof of the Giambelli identity and of Jacobi-Trudi identities. For the …
Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutating relation of generalized Schur operators implies Pieri's f…
The Bernstein operators allow to build recursively the Schur functions. We present a recursion formula for k-Schur functions at t=1 based on combinatorial operators that generalize the Bernstein operators. The recursion leads immediately to a combinatorial interpretation for the expansion coefficients of k-Schur functions at t=1 in terms of homogeneous symmetric functions.
We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act simply on the k-Schur functions, thus leading to a…
We prove that the Lam-Shimozono "down operator" on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of "near rectangles" in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood--Richardson coefficients.
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 …
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 prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.
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…
A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is Δ_L(X;Y)=\det \| x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space M_L is the space spanned by all partial derivatives of Δ_L(X;Y). The ``shift operators'', which are particular partial symmetric derivative operators are very useful in the comprehension of the structure of the M_L spaces. We describe here how a Schur function partial derivative operator acts on l…
We use Young's raising operators to give short and uniform proofs of several well known results about Schur polynomials and symmetric functions, starting from the Jacobi-Trudi identity.
For any Schur function $s_ν$, the associated {\em delta operator} $Δ'_{s_ν}$ is a linear operator on the ring of symmetric functions which has the modified Macdonald polynomials as an eigenbasis. When $ν= (1^{n-1})$ is a column of length $n-1$, the symmetric function $Δ'_{e_{n-1}} e_n$ appears in the Shuffle Theorem of Carlsson-Mellit. More generally, when $ν= (1^{k-1})$ is any column the polynomial $Δ'_{e_{k-1}} e_n$ is the symmetric function side of the Delta Conjecture of Haglund-Remmel-Wilson. We give an expansion of $ωΔ'_{s_ν} e_n$ at $t = 0$ in the dual Hall-Littlewood basis for any part…
Given a partition $\la = (\la_1, \la_2, ... \la_k)$, let $\la^{rc} = (\la_2-1, \la_3-1, ... \la_k-1)$. It is easily seen that the diagram $\la\slash \la^{rc}$ is connected and has no $2 \times 2$ subdiagrams which we shall refer to as a ribbon. To each ribbon $R$, we associate a symmetric function operator $S^R$. We may define the major index of a ribbon $maj(R)$ to be the major index of any permutation that fits the ribbon. This paper is concerned with the operator $H_{1^k}^q = \sum_R q^{maj(R)} S^R$ where the sum is over all $2^{k-1}$ ribbons of size $k$. We show here that $H_{1^k}^q$ has tr…
We give a simple formula for some determinants, and an analogous formula for pfaffians, both of which are polynomial identities. The second involve some expressions that interpolate between determinants and pfaffians. We give several tableau formulas for difference operators in Types A and C, as well as other operators that appear in the "enriched", or "back-stable" Schubert polynomials. There are also tableau formulas for the enriched Schubert polynomials for vexillary and $321$-avoiding permutations.
We use Young's raising operators to derive a Pieri rule for the ring generated by the indeterminates $h_{r,s}$ given in Macdonald's 9th Variation of the Schur functions. Under an appropriate specialisation of $h_{r,s}$, we derive the Pieri rule for the ring $\La(a)$ of double symmetric functions, which has a basis consisting of the double Schur functions. Together with a suitable interpretation of the Jacobi--Trudi identity, our Pieri rule allows us to obtain a new proof of a rule to calculate the Littlewood--Richardson polynomials, which gives a multiplication rule for the double Schur functi…
The Bernstein vertex operators, which can be used to build recursively the Schur functions, are extended to superspace. Four families of super vertex operators are defined, corresponding to the four natural families of Schur functions in superspace. Combinatorial proofs that the super Bernstein vertex operators indeed build the Schur functions in superspace recursively are provided. We briefly mention a possible realization, in terms of symmetric functions in superspace, of the super-KP hierarchy, where the tau-function naturally expands in one of the super-Schur bases.
We study the correlation functions of the Pfaffian Schur process. Borodin and Rains [J. Stat. Phys. 121 (2005), 291-317] introduced the Pfaffian Schur process and derived its correlation functions using a Pfaffian analogue of the Eynard-Mehta theorem. We present here an alternative derivation of the correlation functions using Macdonald difference operators.
Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so prov…
We define crystal operators on semistandard shifted tableaux, giving a new proof that Schur $P$-functions are Schur positive. We define a queer crystal operator to construct a connected queer crystal on semistandard shifted tableaux of a given shape, providing a new proof that products of Schur $P$-functions are Schur $P$-positive. We also give a rectification map from shifted tableaux to Young tableaux that commutes with the crystal operators and provides a dual algorithm to shifted insertion.
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…
