Symplectic $Q$-functions are a symplectic analogue of Schur $Q$-functions and defined as the $t=-1$ specialization of Hall--Littlewood functions associated with the root system of type $C$. In this paper we prove that symplectic $Q$-functions share many of the properties of Schur $Q$-functions, such as a tableau description and a Pieri-type rule. And we present some positivity conjectures, including the positivity conjecture of structure constants for symplectic $P$-functions. We conclude by giving a tableau description of factorial symplectic $Q$-functions.
We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.
By considering the specialisation $s_λ(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $λ$ in terms of two properties of the boxes in the diagram for $λ$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $λ$-tableaux.
A randomisation of the Berele insertion algorithm is proposed, where the insertion of a letter to a symplectic Young tableau leads to a distribution over the set of symplectic Young tableaux. Berele's algorithm provides a bijection between words from an alphabet and a symplectic Young tableau along with a recording oscillating tableau. The randomised version of the algorithm is achieved by introducing a parameter $0 < q < 1$. The classic Berele algorithm corresponds to letting the parameter $q \to 0$. The new version provides a probabilistic framework that allows to prove Littlewood-type ident…
We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.
We present some corollaries to a symplectic primed shifted tableaux version of Tokuyama's identity expressed in terms of other combinatorial constructs, namely generalised $U$-turn alternating sign matrices and strict symplectic Gelfand-Tsetlin patterns.
We introduce an intermediate family of Laurent polynomials between Schur's $Q$-functions and S. Okada's symplectic $Q$-functions. It can also be regarded as a $Q$-function analogue of Proctor's intermediate symplectic characters, and is named the family of intermediate symplectic $Q$-functions. We also derive a tableau-sum formula and a Józefiak-Pragacz-type Pfaffian formula of the Laurent polynomials.
It is known that the right key of a Kashiwara-Nakashima tableau can be computed using the Lecouvey-Sheats symplectic jeu de taquin. Motivated by Willis' direct way of computing type A right keys, we also give a way of computing symplectic right keys without the use of jeu de taquin.
The descent set of an oscillating (or up-down) tableau is introduced. This descent set plays the same role in the representation theory of the symplectic groups as the descent set of a standard tableau plays in the representation theory of the general linear groups. In particular, we show that the descent set is preserved by Sundaram's correspondence. This gives a direct combinatorial interpretation of the branching rules for the defining representations of the symplectic groups; equivalently, for the Frobenius character of the action of a symmetric group on an isotypic subspace in a tensor po…
An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableau of various shapes $λ$ and contents $μ$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableau that "standardi…
The connection between the generating functions of various sets of tableaux and the appropriate families of quasisymmetric functions is a significant tool to give a direct analytical proof of some advanced bijective results and provide new combinatorial formulas. In this paper we focus on two kinds of type B Littlewood-Richardson coefficients and derive new formulas using weak composition quasisymmetric functions and Chow's quasisymmetric functions. In the type A case these coefficients give the multiplication table for Schur functions, i.e. the generating functions for classical Young tableau…
We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.
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.
We introduce the quasi-key basis of the polynomial ring. We prove this basis contains the quasi-Schur polynomials of of Haglund, Luoto, Mason and van Willigenburg and that stable limits of quasi-key polynomials are quasi-Schur functions, thus giving a lifting of the quasi-Schur basis of quasisymmetric polynomials to the full polynomial ring. We introduce the combinatorial model of Kohnert tableaux, and use this model to prove that key polynomials expand positively in quasi-key polynomials which in turn expand positively in the fundamental slide polynomials introduced earlier by the authors. We…
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are…
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 …
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are…
We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed in a previous paper by Gillespie, Levinson, and Purbhoo. These axioms closely resemble those developed by Stembridge for type A tableau crystals. This axiomatic characterization gives rise to a new method for proving and understanding Schur $Q$-positive expansions in symmetric function theory, just as the Stembridge axiomatic structure provides for ordinary Schur positivity.
Given a Schubert class on $Gr(k,V)$ where $V$ is a symplectic vector space of dimension $2n$, we consider its restriction to the symplectic Grassmannian $SpGr(k,V)$ of isotropic subspaces. Pragacz gave tableau formulae for positively computing the expansion of these $H^*(Gr(k,V))$ classes into Schubert classes of the target when $k=n$, which corresponds to expanding Schur polynomials into $Q$-Schur polynomials. Coskun described an algorithm for their expansion when $k\leq n$. We give a puzzle-based formula for these expansions, while extending them to equivariant cohomology. We make use of a n…
We construct colored lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang--Baxter equation that allows us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use …
Over the past years, major attention has been drawn to the question of identifying Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions. The set of arc permutations, i.e. the set of permutations $π$ in $S_n$ such that for any $1\leq j \leq n$, $\{π(1),π(2),\dots,π(j)\}$ is an interval in $\mathbb{Z}_n$ is one of the most noticeable examples. This paper introduces a new type B extension of Schur-positivity to signed permutations based on Chow's quasisymmetric functions and gener…
We present explicit formulas for the Macdonald polynomials of types $C_n$ and $D_n$ in the one-row case. In view of the combinatorial structure, we call them "tableau formulas". For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced ${}_{12}W_{11}$ series. We remark that the correlation functions of the deformed $\mathcal{W}$ algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables.
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positi…
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' + …
This paper contains a partial answer to the open problem 3.11 of \cite{[H2008]}. That is to find an explicit bijection on Schröder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over $m$-Schröder paths with a fix number of diagonal steps into Schur functions in the variables $q$ and $t$. Some results have been generalized to parking functions, and some bijections were made with standard Young tableaux giving way to partial combinatorial formulas in the basis $s_μ(q,t)s_λ(X)$ for $\nabla(e_n)$ (respectively, $\nabla^m(e_n)$), when $μ$ and $λ…
We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions -- Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.
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 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 …
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 present two symmetric function operators $H_3^{qt}$ and $H_4^{qt}$ that have the property $H_{3}^{qt} H_{(2^a1^b)}[X;q,t] = H_{(32^a1^b)}[X;q,t]$ and $H_4^{qt} H_{(2^a1^b)}[X;q,t] = H_{(42^a1^b)}[X;q,t]$. These operators are generalizations of the analogous operator $H_2^{qt}$ and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, $a_μ(T)$ and $b_μ(T)$, on standard tableaux such that the $q,t$ Kostka polynomials are given by the sum over standard tableaux of shape $\la$, $K_{\laμ}(q,t) = \sum_T t^{a_μ(T)} q^{b_μ(T)}$ for the case when when $μ$ is…
The expectation of the descent number of a random Young tableau of a fixed shape is given, and concentration around the mean is shown. This result is generalized to the major index and to other descent functions. The proof combines probabilistic arguments together with combinatorial character theory. Connections with Hecke algebras are mentioned.
Shuffles are n-multipermutations with suit multiplicities given by a subset R of {1,..,n-1}. Their inverses are ordered partitions of {1,..,n} whose block sizes derive from R. These "R-permutations" depict the min length coset reps for the quotient of S_n by the parabolic subgroup W_J, with J the complement of R. We refer to those that blockwise avoid the pattern 312 as "312-avoiding R-permutations" and define the "parabolic R-Catalan number" to be the number of them. Let lambda be a partition of N with at most n parts whose set of shape column lengths less than n is R. We show that the number…
In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood-Richardson coefficients in torus-equivariant $K$-theory of Grassmannians. We then studied the genomic Schur function $U_λ$, a generating function for such tableaux, showing that it is non-trivially a symmetric function, although generally not Schur-positive. Here we show that $U_λ$ is, however, positive in the basis of fundamental quasisymmetric functions. We give a positive combinatorial formula for this expansion in terms of gapless increasing tableaux; this is, more…
We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kraśkiewicz and developed by Lam and Serrano. We also introduce generalized Littlewood--Richardson coefficients for products of Schur's symmetric functions and give combinatorial descriptions in terms of tableau words. The insertion algorithms play central roles for proofs. A new description of shifted Littlewood--Richardson coefficients is given in terms of semistandard increasing decomposition tableaux. We show that a "big" Schur …
Ikeda-Mihalcea-Naruse's double Schubert polynomials represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined in terms of (flagged) marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-f…
We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall-Littlewood functions using Young's raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for Hall-Littlewood functions, theta polynomials, and related Stanley symmetric functions. Finally, we introduce the notion of a skew element w …
A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob--Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to type B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of type B and D and quasisymmetric functions and noncommutative symmetric functions of type B and D. Other applications are also provided.
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 …
The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_λ^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young l…
Let $Λ$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_λ[X/(1-t)]\}_{λ_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_λ^{(k)}[X;t]\}_{λ_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_λ^{(k)}[X;t]$ form a basis for $V_k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but wo…
The k-Schur functions were first introduced by Lapointe, Lascoux and Morse (2003) in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono (2010) as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type A by Lam (2008), and, at t …
We consider the expansion of the square of a complete homogeneous function $h_λ$, or of an elementary symmetric function $e_λ$, in the basis of Schur functions. This square also decomposes into two plethysms, $s_2[h_λ]$ and $s_{11}[h_λ]$ (resp. $s_2[h_λ]$ and $s_{11}[h_λ]$), which are called its symmetric and anti-symmetric parts, respectively. We define a sign statistic on the set of tableaux that index the Schur functions appearing in the square of those symmetric functions. This sign statistic allows to determine to which plethysm each Schur function contributes. We use mainly combinatorial…
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…
