Masao Ishikawa, Soichi Okada, Hiroyuki Tagawa, Jiang Zeng
We present several generalizations of Cauchy's determinant and Schur's Pfaffian by considering matrices whose entries involve some generalized Vandermonde determinants. Special cases of our formulae include previuos formulae due to S.Okada and T. Sundquist. As an application, we give a relation for the Littlewood--Richardson coefficients involving a rectangular partition.
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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.
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.
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。
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…
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。
For any complex number $α$ and any even-size skew-symmetric matrix $B$, we define a generalization $\pfaα(B)$ of the pfaffian $\pf(B)$ which we call the $α$-pfaffian. The $α$-pfaffian is a pfaffian analogue of the $α$-determinant. It gives the pfaffian at $α=-1$. We give some formulas for $α$-pfaffians and study the positivity. Further we define point processes determined by the $α$-pfaffian. Also we provide a linear algebraic proof of the explicit pfaffian expression for the correlation function of the shifted Schur measure.
We present a Pfaffian identity involving elliptic functions, whose rational limit gives a generalization of Schur's Pfaffian identity for Pf ((x_j - x_i)/(x_j + x_i)). This identity is regarded as a Pfaffian counterpart of Frobenius identity, which is an elliptic generalization of Cauchy's determinant identity for det (1/ (x_i + x_j)).
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.
We study pfaffian analogues of immanants, which we call pfaffinants. Our main object is the TL-pfaffinants which are analogues of Rhoades and Skandera's TL-immanants. We show that TL-pfaffinants are positive when applied to planar networks and explain how to decompose products of complementary pfaffians in terms of TL-pfaffinants. We conjecture in addition that TL-pfaffinants have positivity properties related to Schur Q-functions.
We introduce interpolation analogues of Schur Q-functions - the multiparameter Schur Q-functions. We obtain for them several results: a combinatorial formula, generating functions for one-row and two-rows functions, vanishing and characterization properties, a Pieri-type formula, a Nimmo-type formula (a relation of two Pfaffians), a Giambelli-Schur-type Pfaffian formula, a determinantal formula for the transition coefficients between multiparameter Schur Q-functions with different parameters. We write an explicit Pfaffian expression for the dimension of skew shifted Young diagram. This paper i…
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.
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 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.
We present an identity which can be regarded as a Pfaffian-Hafnian analogue of Borchardt's identity and as a generalization of Schur's identity. We give a proof using the complex analysis.
In this article, we use Lindström Gessel Viennot Lemma to give a short, combinatorial, visualizable proof of the identity of Schur polynomials -- the sum of monomials of Young tableaux equals to the quotient of determinants. As a by-product, we have a proof of Vandermonde determinant without words. We also prove the cauchy identity. In the remarks, we discuss factorial Schur polynomials, dual Cauchy identity and the relation beteen Newton interpolation formula.
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.
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。
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。
We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of "free boundary states". For the case of one free boundary, our approach yields a new pro…
More minor summation formulae
本文推导出同时推广Ishikawa和Wakayama以及Byun最近小项求和公式的行列式-佩尔数公式，基于斜对称矩阵与秩-1矩阵和的行列式分解公式，并应用于斜Schur函数的Cauchy型恒等式。
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。
Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials
本研究针对$q$-Whittaker与斜Schur多项式的受限柯西和，借助可积概率中的$q$-Whittaker测度、周期Schur测度及其Fredholm行列式公式，证明了二者间的恒等式，拓展了对称函数柯西恒等式的相关结论。
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…
A Littlewood-type identity for Robbins polynomials
该研究推广了利特伍德恒等式，将舒尔多项式替换为修正罗宾斯多项式，通过 Pfaffian 公式关联六顶点模型配分函数，拓展了交替符号矩阵的相关计数理论。
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 examine and present new combinatorics for the Schur polynomials from the viewpoint of quantum integrability. We introduce and analyze an integrable six-vertex model which can be viewed as a certain degeneration model from a t-deformed boson model. By a detailed analysis of the wavefunction from the quantum inverse scattering method, we present a novel combinatorial formula which expresses the Schur polynomials by using an additional parameter, which is in the same sense but different from the Tokuyama formula. We also give an algebraic analytic proof for the Cauchy identity and make applica…
Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $\mathsf{F}_λ$ arise in the context of $\mathfrak{sl}(2)$ higher spin six vertex models, and are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two $\mathsf{F}_λ$'s as a determinant. The determinant is of Izergin-Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang-Baxter…
We attempt to explain the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations of Heisenberg algebras. The resulting framework that we describe is a generalization of the classical Boson-Fermion correspondence, from which Schur functions arise. Our work can be used to understand Hall-Littlewood polynomials, Macdonald polynomials and Lascoux, Leclerc and Thibon's ribbon functions, together with other new families of symmetric functions…
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