This paper proves an identity between flagged Schur polynomials, giving a duality between row flags and column flags. This identity generalises both the binomial determinant duality theorem due to Gessel and Viennot and the symmetric function duality theorem due to Aitken. As corollaries we obtain the lifts of the binomial determinant duality theorem to $q$-binomial coefficients and to symmetric polynomials. Our method is a path counting argument on a novel lattice generalising that used by Gessel and Viennot.
We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations.
Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and give…
Schur polynomials are special cases of Schubert polynomials, which in turn are special cases of dual characters of flagged Weyl modules. The principal specialization of Schur and Schubert polynomials has a long history, with Macdonald famously expressing the principal specialization of any Schubert polynomial in terms of reduced words. We study the principal specialization of dual characters of flagged Weyl~modules. Our result yields an alternative proof of a conjecture of Stanley about the principal specialization of Schubert polynomials, originally proved by Weigandt.
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…
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…
Characterization of $\mathfrak{k}$-highest weight and $\widehat{\mathfrak{g}}$-dominant tableaux via $1$-$0$-slack recording tableaux in the quantum Littlewood-Richardson rule and lattice points in flagged hive polytopes
Center for Combinatorics, LPMC Nankai University（南开大学组合数学中心，LPMC）；School of Mathematical Sciences, Tianjin University of Technology（天津工业大学理学院）；Naval University of Engineering（海军工程大学）
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…
Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in $x=(x_1,x_2,\dots)$ with additional parameters $t=(t_1,t_2,\dots)$. The refined dual stable Grothendieck polynomials are defined as a generating function for reverse plane partitions of a given shape. They interpolate between Schur functions and dual stable Grothendieck polynomials introduced by Lam and Pylyavskyy in 2007. Flagged refined dual stable Grothendieck polynomials are a more refined version of refined dual stable Grothendieck polynomials, where lower and…
We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.
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 the theory of $(\mathcal{P},ρ)$-partitions, depending on a poset $\mathcal{P}$ and a map $ρ$ from $\mathcal{P}$ to positive integers. The generating function $\mathfrak{F}_{\mathcal{P},ρ}$ of $(\mathcal{P},ρ)$-partitions is a polynomial that, when the images of $ρ$ tend to infinity, tends to Stanley's generating function $F_{\mathcal{P}}$ of $\mathcal{P}$-partitions. Analogous to Stanley's fundamental theorem for $\mathcal{P}$-partitions, we show that the set of $(\mathcal{P},ρ)$-partitions decomposes as a disjoint union of $(\mathcal{L},ρ)$-partitions where $\mathcal{L}$ runs ove…
A remarkable result of Stanley shows that the set of maximal chains in the non-crossing partition lattice of type $A$ is Schur-positive, where descents are defined by a distinguished edge labeling. A bijection between these chains and labeled trees was presented by Goulden and Yong. Using Adin-Roichman's variant of Björner's $EL$-labeling, we show that the subset of maximal chains in the non-crossing partition lattice of type $A$, whose underlying tree is a convex caterpillar, is Schur-positive.
Let $1\leq k \leq n$ and let $X_n = (x_1, \dots, x_n)$ be a list of $n$ variables. The {\em Boolean product polynomial} $B_{n,k}(X_n)$ is the product of the linear forms $\sum_{i \in S} x_i$ where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots, n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call {\em Chern plethysm}. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a com…
This paper proves a combinatorial rule expressing the product $s_τ(s_{λ/μ} \circ p_r)$ of a Schur function and the plethysm of a skew Schur function with a power sum symmetric function as an integral linear combination of Schur functions. This generalizes the SXP rule for the plethysm $s_λ\circ p_r$. Each step in the proof uses either an explicit bijection or a sign-reversing involution. The proof is inspired by an earlier proof of the SXP rule due to Remmel and Shimozono, A simple proof of the Littlewood--Richardson rule and applications, Discrete Mathematics 193 (1998) 257--266. The connecti…
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…
Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of non-crossing trees is a free ternary magma with one generator and this duality is an instance of a duality that is defined in any such magma. Any two free ternary magmas with one generator are isomorphic via a unique isomorphism that we call the structural bijection. Besides the set of non-crossing trees we also consider as free ternary magmas with one generator the s…
We conjecture an explicit formula for the image of a tensor product of Kirillov-Reshetikhin crystals $\bigotimes_{i=1}^m B^{1, s_i}$ under the Kirillov-Schilling-Shimozono bijection. Our conjectured formula is piecewise-linear, where the shapes are given by the tropicalization of cylindric loop Schur functions and the riggings are given by the tropicalization of loop Schur functions. We prove that our formula changes the riggings by the correct amount based upon the time evolution of the corresponding box-ball system. We show that our formula is correct under the column splitting portion of th…
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…
We exhibit a canonical connection between maximal (0,1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between k-triangulations of the n-gon and k-fans of Dy…
In a recent paper, Ayyer and Behrend present for a wide class of partitions factorizations of Schur polynomials with an even number of variables where half of the variables are the reciprocals of the others into symplectic and/or orthogonal group characters, thereby generalizing results of Ciucu and Krattenthaler for rectangular shapes. Their proofs proceed by manipulations of determinants underlying the characters. The purpose of the current paper is to provide bijective proofs of such factorizations. The quantities involved have known combinatorial interpretations in terms of Gelfand-Tsetlin…
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…
We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we concern with the case of to the classical flag manifold of Lie type A and discuss related combinatorial objects: flagged Schur polynomi…
We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics `number of E steps shared with B' and `number of E steps shared with T' have a symmetric joint distribution. To do so, we give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is inde…
