Bilinear identities on Schur symmetric functions

Bijective proofs for Schur function identities which imply Dodgson's condensation formula and Plücker relations

We present a ``method'' for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi--Trudi identity. We illustrate this ``method'' by generalizing a bijective construction (which was first used by Goulden) to a class of Schur function identities, from which we shall obtain bijective proofs for Dodgson's condensation formula, Plücker relations and a recent identity of the second author.

Pfaffian Formulas and Schur Q-Function Identities

We establish Pfaffian analogues of the Cauchy--Binet formula and the Ishikawa--Wakayama minor-summation formula. Each of these Pfaffian analogues expresses a sum of products of subpfaffians of two skew-symmetric matrices in terms of a single Pfaffian. By using these Pfaffian formulas we give new transparent proofs to several identities for Schur Q23 pa-functions.

Bilinear expansion of Schur functions in Schur $Q$-functions: a fermionic approach

An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of products of either charged or neutral fermionic creation and annihilation operators, Wick's theorem and a factorization identity for VEV's of products of two mutually anticommuting sets of neutral fermionic operators.

Elementary proofs of identities for Schur functions and plane partitions

Elementary proofs are given for sums of Schur functions over partitions into at most n parts each less than or equal to m for which i) all parts are even, ii) all parts of the conjugate partition are even. Also, an elementary proof of a recent result of Ishikawa and Wakayama is given.

Overlap Identities for Littlewood-Schur Functions

Our results revolve around a new operation on partitions, which we call overlap. We prove two overlap identities for so-called Littlewood-Schur functions. Littlewood-Schur functions are a generalization of Schur functions, whose study was introduced by Littlewood. More concretely, the Littlewood-Schur function $LS_λ(X; Y)$ indexed by the partition $λ$ is a polynomial in the variables $X \cup Y$ that is symmetric in both $X$ and $Y$ separately. The first overlap identity represents $LS _λ(X; Y)$ as a sum over subsets of $X$, while the second overlap identity essentially represents $LS_λ(X; Y)$ …

A combinatorial proof of the Rogers-Ramanujan and Schur identities

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.

Determinantal and Pfaffian identities for ninth variation skew Schur functions and Q-functions

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalizing those of Josefiak-Pragacz, Nimmo, and most recently Okada. As a preamble to this we present a parallel develo…

A GL-Equivariant Complex Inducing Character Identities for Schur Modules

In this paper we construct a GL-equivariant complex of Schur modules over a ring of positive characteristic that can be used to deduce classical alternating sum identities for Schur polynomials. This complex globalizes to a complex of vector bundles and can also be used to give an explicit construction of an exact sequence predicted by work of Grayson involving Adams operations identities on the algebraic K-theory of a given scheme $X$. The more general complex gives an explicit construction that reproves the aforementioned Adams operations identities in full generality.

柱形Schur函数的有界Littlewood恒等式

Bounded Littlewood identities for cylindric Schur functions

本文提出有界Littlewood恒等式的仿射类似物，给出柱形Schur函数求和的行列式公式，并揭示柱形标准杨表与r-非交叉、s-非嵌套匹配间的新联系。

Schur function identities and the number of perfect matchings of holey Aztec rectangles

We compute the number of perfect matchings of an $M\times N$ Aztec rectangle where $|N-M|$ vertices have been removed along a line. A particular case solves a problem posed by Propp. Our enumeration results follow from certain identities for Schur functions, which are established by the combinatorics of nonintersecting lattice paths.

Comultiplication rules for the double Schur functions and Cauchy identities

The double Schur functions form a distinguished basis of the ring Λ(x||a) which is a multiparameter generalization of the ring of symmetric functions Λ(x). The canonical comultiplication on Λ(x) is extended to Λ(x||a) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood-Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the ide…

Generalized Sylvester Formulas and skew Giambelli Identities

We obtain a common generalization of two types of Sylvester formulas for compound determinants and its Pfaffian analogue. As applications, we give generalizations of the Giambelli identity to skew Schur functions and the Schur identity to Schur's skew $Q$-functions.

Bijective proofs for Schur function identities

Gurevich, Pyatov and Saponov recently stated an expansion for the product of two Schur functions and gave a proof based on the Pluecker relations. Here we show that this identity is in fact a special case of a quite general Schur function identity, which was stated and proved in a paper by Fulmek and Kleber, where it was used to prove bijectively Dodgsons condensation formula and the Pluecker relations, but was not paid further attention: So we take the opportunity to make obvious the range of applicability of this identity by giving concrete examples, accompanied by many graphical illustratio…

Identities for Schur-type determinants and pfaffians

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.

Schur function identities arising from the basic representation of $A^{(2)}_{2}$

A Lie theoretic interpretation is given for some formulas of Schur functions and Schur $Q$-functions. Two realizations of the basic representation of the Lie algebra $A^{(2)}_2$ are considered; one is on the fermionic Fock space and the other is on the bosonic polynomial space. Via the boson-fermion correspondence, simple relations of the vacuum expectation values of fermions turn out to be algebraic relations of Schur functions.

Factorial supersymmetric skew Schur functions and ninth variation determinantal identities

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 …

Interlacing networks: birational RSK, the octahedron recurrence, and Schur function identities

Motivated by the problem of giving a bijective proof of the fact that the birational RSK correspondence satisfies the octahedron recurrence, we define interlacing networks, which are certain planar directed networks with a rigid structure of sources and sinks. We describe an involution that swaps paths in these networks and leads to Plücker-like three-term relations among path weights. We show that indeed these relations follow from the Plücker relations in the Grassmannian together with some simple rank properties of the matrices corresponding to our interlacing networks. The space of matrice…

斜辛 Schur 函数与斜正交 Schur 函数

Skew Symplectic and Orthogonal Schur Functions

本文利用辛与正交 Schur 函数的顶点算子表示定义并刻画斜辛、斜正交 Schur 函数，证明其分支规则并导出 Jacobi–Trudi、Gelfand–Tsetlin 与 Cauchy 型结果，实现 A、C、D 型的统一构造。

Orthosymplectic Cauchy identities

We give bijective proofs of orthosymplectic analogues of the Cauchy identity and dual Cauchy identity for orthosymplectic Schur functions. To do so, we present two insertion algorithms; these are orthosymplectic versions of Berele's symplectic insertion algorithms, which were used by Sundaram to give bijective proofs of Cauchy identities for symplectic Schur functions.

重访Jacobi-Trudi恒等式：通过BGG范畴 $\mathcal{O}$

Revisiting Jacobi-Trudi identities via the BGG category $\mathcal{O}$

本文通过BGG范畴O中的张量积重数重新证明斜Schur多项式的Jacobi-Trudi恒等式，并确立截断展开的Schur正性，同时解释系数为张量积重数。

Pfaffian-Toeplitz恒等式、Schur正性与Baxter多项式的q-对数凸性

Pfaffian--Toeplitz identities, Schur positivity, and the $q$-log-convexity of Baxter polynomials

该研究利用Pfaffian与Toeplitz相关公式，证明Baxter多项式序列具q-对数凸性，其q-精细Baxter数阵列行与列均为q-对数凹，相关结果可推广至d-Hoggatt数的q-模拟。

Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs

Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.

与 Lebesgue、Schur 和 Capparelli 的分拆定理相关的一些 $q$-超几何恒等式

Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli

本文构造含参数 $L,M$ 的三变量多项式恒等式，统一推广并在极限下导出 Schur 与 Lebesgue 关键恒等式，还给出 Alladi–Schur 精细化及连通 Euler、Lebesgue、Capparelli 定理的新无限层级。

A–D型经典群特征标的第九变体与Littlewood恒等式

Ninth variation of classical group characters of type A-D and Littlewood identities

本文推广经典李群特征标与有理Schur函数，用任意多项式序列替代阶乘幂，推导Littlewood型恒等式并获得新的Jacobi–Trudi与Nägelsbach–Kostka恒等式。

Rogers-Ramanujan恒等式与Cauchy恒等式

The Rogers-Ramanujan Identities and Cauchy's Identity

本文利用Schur函数的Cauchy恒等式对Rogers-Ramanujan恒等式进行了研究。

Vertex operators arising from Jacobi-Trudi identities

We give an interpretation of the boson-fermion correspondence as a direct consequence of Jacobi-Trudi identity. This viewpoint enables us to construct from a generalized version of the Jacobi-Trudi identity the action of Clifford algebra on polynomial algebras that arrives as analogues of the algebra of symmetric functions. A generalized Giambelli identity is also proved to follow from that identity. As applications, we obtain explicit formulas for vertex operators corresponding to characters of the classical Lie algebras, shifted Schur functions, and generalized Schur symmetric functions asso…

Frobenius's last proof

Around about 1917, Issai Schur rediscovered the Rogers-Ramanujan identities, and proved a system of polynomial identities that imply them. Schur wrote that Georg Frobenius (his former advisor) had shown him a simple, direct proof of these polynomial identities. Schur did not see fit to reveal Frobenius's proof, preferring his own rather complicated proof. But it is easy enough to guess what this `simple, direct' proof must have been. As Frobenius died in 1917, we may call this `Frobenius's last proof'.

Identities on Factorial Grothendieck Polynomials

Gustafson and Milne proved an identity on the Schur function indexed by a partition of the form $(λ_1-n+k,λ_2-n+k,\ldots,λ_k-n+k)$. On the other hand, Fehér, Némethi and Rimányi found an identity on the Schur function indexed by a partition of the form $(m-k,\ldots,m-k, λ_1,\ldots,λ_k)$. Fehér, Némethi and Rimányi gave a geometric explanation of their identity, and they raised the question of finding a combinatorial proof. In this paper, we establish a Gustafson-Milne type identity as well as a Fehér-Némethi-Rimányi type identity for factorial Grothendieck polynomials. Specializing a factorial…

Refined dual Grothendieck polynomials, integrability, and the Schur measure

We construct a vertex model whose partition function is a refined dual Grothendieck polynomial, where the states are interpreted as nonintersecting lattice paths. Using this, we show refined dual Grothendieck polynomials are multi-Schur functions and give a number of identities, including a Littlewood and Cauchy(-Littlewood) identity. We then refine Yeliussizov's connection between dual Grothendieck polynomials and the last passage percolation (LPP) stochastic process discussed by Johansson. By refining algebraic techniques of Johansson, we show Jacobi-Trudi formulas for skew refined dual Grot…

展开K-理论Schur Q-函数

Expanding K-theoretic Schur Q-functions

该研究推导了K-理论Schur P/Q函数相关恒等式，证明了Lewis等人猜想的Q函数按P函数展开的公式，推广至斜与对偶版本，还证明移位斜Cauchy恒等式，并探讨对偶函数的猜想公式及其基性质推论。

Notes about KP/BKP correspondence

I present a set of remarks related to joint works \cite{paper1},\cite{paper2},\cite{paper3},\cite{MMNO}. These are remarks about polynomials solutions and vertex operators, eigenproblem for polynomials and a remark related to the the conjecture of Alexandrov and Mironov, Morozov about the ratios of the projective Schur functions. New results on the bilinear relations between characters of the symmetric and of the Sergeev group and on bilinear relations between skew Schur and projective Schur functions and also between shifted Schur and projective Schur functions are added. Certain new matrix m…

加权双线性恒等式与Apéry型多项式的超同余

Weighted bilinear identities and supercongruences for Apéry-like polynomials

本文为两类Apéry型多项式建立加权双线性求和恒等式，结合端点值同余式得到模p^3和p^4的超同余式及特殊求值，并证实了Sun的一个猜想。

A Generalization of Schur's $P$- and $Q$-Functions

We introduce and study a generalization of Schur's $P$-/$Q$-functions associated to a polynomial sequence, which can be viewed as ``Macdonald's ninth variation'' for $P$-/$Q$-functions. This variation includes as special cases Schur's $P$-/$Q$-functions, Ivanov's factorial $P$-/$Q$-functions and the $t=-1$ specialization of Hall--Littlewood functions associated to the classical root systems. We establish several identities and properties such as generalizations of Schur's original definition of Schur's $Q$-functions, Cauchy-type identity, Józefiak--Pragacz--Nimmo formula for skew $Q$-functions…

Schur Positivity and the $q$-Log-convexity of the Narayana Polynomials

Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the $q$-log-convexity of the Narayana polynomials, and a proof of the second conjecture that the Narayana transformation preserves the log-convexity. Based on a formula of Bränd$\mathrm{\acute{e}}$n which expresses the $q$-Narayana numbers as the specializations of Schur functions, we derive several symmetric function identities using the Littlewood-Richardson rule for the product of Schur functions, and obtain the strong $q$-log-convexity of the Narayana pol…

Generalized Schur function determinants using the Bazin identity

In the literature there are several determinant formulas for Schur functions: the Jacobi-Trudi formula, the dual Jacobi-Trudi formula, the Giambelli formula, the Lascoux-Pragacz formula, and the Hamel-Goulden formula, where the Hamel-Goulden formula implies the others. In this paper we use an identity proved by Bazin in 1851 to derive determinant identities involving Macdonald's 9th variation of Schur functions. As an application we prove a determinant identity for factorial Schur functions conjectured by Morales, Pak, and Panova. We also obtain a generalization of the Hamel-Goulden formula, w…

Using Grassmann calculus in combinatorics: Lindström-Gessel-Viennot lemma and Schur functions

Grassmann (or anti-commuting) variables are extensively used in theoretical physics. In this paper we use Grassmann variable calculus to give new proofs of celebrated combinatorial identities such as the Lindström-Gessel-Viennot formula for graphs with cycles and the Jacobi-Trudi identity. Moreover, we define a one parameter extension of Schur polynomials that obey a natural convolution identity.

Schur $Q$-函数的Pfaffian表述

Pfaffian Formulation of Schur's $Q$-functions

本研究提出可拓展Schur $Q$-函数索引范围的Pfaffian公式，开发了对应的分解证明技术，实现了用简单代数方法证明该函数的若干恒等式，完善了其构造体系。

Differential operators on Schur and Schubert polynomials

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 …

Vertex operators, Weyl determinant formulae and Littlewood duality

Vertex operator realizations of symplectic and orthogonal Schur functions are studied and expanded. New proofs of determinant identities of irreducible characters for the symplectic and orthogonal groups are given. We also give a new proof of the duality between the universal orthogonal and symplectic Schur functions using vertex operators.

Bilinear expansions of lattices of KP $τ$-functions in BKP $τ$-functions: a fermionic approach

We derive a bilinear expansion expressing elements of a lattice of KP $τ$-functions, labelled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP $τ$-functions, labelled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur $Q$-functions. It is deduced using the representations of KP and BKP $τ$-functions as vacuum expectation values (VEV's) of products of fermionic operators of charged and neutral type, respectively. The lattice is ge…

A combinatorial proof of a plethystic Murnaghan--Nakayama rule

This article gives a combinatorial proof of a plethystic generalization of the Murnaghan--Nakayama rule. The main result expresses the product of a Schur function with the plethysm $p_r \circ h_n$ as an integral linear combination of Schur functions. The proof uses a sign-reversing involution on sequences of bead moves on James' abacus, inspired by the arguments in N. Loehr, Abacus proofs of Schur function identities, SIAM J. Discrete Math. 24 (2010), 1356-1370.

Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions

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…

多变量阿尔 - 萨拉姆 - 卡尔itz 多项式的双线性生成函数及其应用

Bilinear generating functions of the multivariable Al-Salam-Carlitz polynomials and applications

研究多变量阿尔 - 萨拉姆 - 卡尔itz 多项式，通过比较系数法建立新生成函数，其含特殊公式，还给出相关新\(q -\)级数恒等式。

Lee、Rains 与 Warnaar 猜想的一些 Littlewood 恒等式的证明

Proof of some Littlewood identities conjectured by Lee, Rains and Warnaar

本文证明了 Lee、Rains 和 Warnaar 在 Macdonald 情形下猜想的一对关于 Schur 函数的新 Littlewood 恒等式，其中求和对空 2-core 分拆进行，并由此得到一个符合 Littlewood 原始公式精神的新恒等式。

(GL_k x S_n)-modules and nabla of hook-indexed Schur functions

The aim of this paper is to describe structural properties of spaces of diagonal rectangular harmonic polynomials in several sets (say $k$) of $n$ variables, both as $GL_k$-modules and $S_n$-modules. We construct explicit such modules associated to any hook shape partitions. For the two sets of variables case, we conjecture that the associated graded Frobenius characteristic corresponds to the effect of the operator Nabla on the corresponding hook-indexed Schur function, up to a usual renormalization. We prove identities that give indirect support to this conjecture, and show that its restrict…

Duality and deformations of stable Grothendieck polynomials

Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials, and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi type identities, and associated Fomin-Greene operators.

使用Loehr的标记算盘证明plethystic Murnaghan-Nakayama规则

Proof of the plethystic Murnaghan-Nakayama rule using Loehr's labelled abacus

本文利用Loehr提出的标记算盘，为plethystic Murnaghan-Nakayama规则提供了一种简短且完全组合的证明，该规则描述了Schur函数与形如$p_r\circ h_m$的plethysm乘积如何分解为Schur函数之和。

Schur's partition theorem and mixed mock modular forms

We study families of partitions with gap conditions that were introduced by Schur and Andrews, and describe their fundamental connections to combinatorial q-series and automorphic forms. In particular, we show that the generating functions for these families naturally lead to deep identities for theta functions and Hickerson's universal mock theta function, which provides a very general answer to Andrews' Conjecture on the modularity of the Schur-type generating function. Furthermore, we also complete the second part of Andrews' speculation by determining the asymptotic behavior of these funct…

仿射根系Macdonald恒等式的一些组合解释及Nekrasov--Okounkov型公式

Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas

通过整数向量与双无限词表示的整数分拆之间的关联，给出仿射根系Macdonald恒等式的组合解释，并推导出各类型的q-Nekrasov--Okounkov公式。

The Hopf algebra of odd symmetric functions

We consider a q-analogue of the standard bilinear form on the commutative ring of symmetric functions. The q=-1 case leads to a Z-graded Hopf superalgebra which we call the algebra of odd symmetric functions. In the odd setting we describe counterparts of the elementary and complete symmetric functions, power sums, Schur functions, and combinatorial interpretations of associated change of basis relations.

Plethysms of Chromatic and Tutte Symmetric Functions

Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function plethysms is a major open question. In this paper, we introduce a graph-theoretic interpretation for any plethysm based on the chromatic symmetric function. We use this interpretation to give simple proofs of new and previously known plethystic identities, as well as chromatic symmetric function identities.

右钥匙的新刻画，以及t=0处的m对称舒尔函数

A new characterization of right keys, and the $m$-symmetric Schur functions at $t=0$

该研究刻画右钥匙表，发展m对称函数环的舒尔函数在t=0处的组合理论，证明相关柯西恒等式，关联m对称舒尔函数与几乎对称舒尔函数并推导行列式公式。

四面体L-算子、张量Schur多项式和q-变形的环初等对称函数

Tetrahedral $L$-operators, tensor Schur polynomials and $q$-deformed loop elementary symmetric functions

本文基于四面体L-算子研究三维分区函数，提出可表示为张量Schur多项式类乘积的分区函数，推导Schur多项式洗牌公式及多个恒等式的统一，并展示其在多物种完全非对称排斥过程稳态中的应用，同时确定q-变形环初等对称函数的显式形式。

顶点算子、无限楔表示和t-Schur测度的相关函数

Vertex operators, infinite wedge representations, and correlation functions of the t-Schur measure

本文通过顶点算子方法研究t-Schur测度，推导其相关函数并证明其确定性，同时揭示其与Schur和Schur-Q理论的联系。