F. Jouhet, J. Zeng
Some new identities for Schur functions are proved. In particular, we settle in the affirmative a recent conjecture of Ishikawa-Wakayama and solve a problem raised by Bressoud.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.
Some new identities for Schur polynomials
本文提出了关于Schur多项式的新恒等式，扩展了Cauchy恒等式的有界和无界广义形式。
We give simple elementary proofs of Bressoud's and Schur's polynomial versions of the Rogers-Ramanujan identities
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 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.
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.
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.
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.
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 $LS_λ(X; Y)$ …
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.
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…
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。
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.
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…
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.
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…
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.
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 …
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…
A series of bilinear identities on the Schur symmetric functions is obtained with the use of Pluecker relations.
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.
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.
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。
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.
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.
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。
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…
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。
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。
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。
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。
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'.
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…
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…
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…
We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kurşungöz. We also present some related polynomial and $q$-series identities.
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。
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 …
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.
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。
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 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…
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.
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…
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。
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.
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。
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.
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。
We investigate the $e$-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular $e$-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott's triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the $e$-positivity and Schur positivity of trees.
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…
Schur-hooks and Bernoulli number recurrences
本文利用 Schur 钩形函数的乘积展开、符号反转对合及 Hoffman 的多重调和级数结果，建立相关对称函数恒等式，并导出 Bernoulli 数的新递推式。
Some combinatorial interpretations of the Macdonald identities for affine root systems
本文利用整数向量与整数划分的联系，通过辛Schur函数和特殊正交Schur函数，给出了七类无限族仿射根系Macdonald恒等式的组合解释，并推导了相关的q-Nekrasov--Okounkov公式。
Subsequently to the author's preceding paper, we give full proofs of some explicit formulas about factorizations of $K$-$k$-Schur functions associated with any multiple $k$-rectangles.
The product $s_μs_ν$ of two Schur functions is one of the most famous examples of a Schur-positive function, i.e. a symmetric function which, when written as a linear combination of Schur functions, has all positive coefficients. We ask when expressions of the form $s_λs_ρ- s_μs_ν$ are Schur-positive. This general question seems to be a difficult one, but a conjecture of Fomin, Fulton, Li and Poon says that it is the case at least when $λ$ and $ρ$ are obtained from $μ$ and $ν$ by redistributing the parts of $μ$ and $ν$ in a specific, yet natural, way. We show that their conjecture is true in s…
Some conjectures on the Schur expansion of Jack polynomials
本文提出Jack多项式Schur展开的正性猜想，并探讨了相关基底中的组合学内容及特殊情形的证明。
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.
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…
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.
In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。