arXiv math/06128282006-12-28math.CO

Non symmetric Cauchy kernels for the classical Groups

Amy M. Fu, Alain Lascoux

英文摘要：

We give non-symmetric versions of the Cauchy kernel and Littlewood's kernels, corresponding to the types $A_n$, $B_n$, $C_n$ and $D_n$, of the classical groups. We show that these new kernels are diagonal in the basis of two families of key polynomials (one of them being Demazure characters) obtained as images of dominant monomials under isobaric divided differences. We define scalar products such that the two families of key polynomials are adjoint to each other.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

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.

2018-09-14math.CO

非对称 Cauchy 核、晶体与末次通过渗流

Non symmetric Cauchy kernel, crystals and last passage percolation

本文借助非对称 Cauchy 核恒等式与兼容晶体基理论的统一 RSK 限制，给出末次通过渗流模型规律的 Demazure 特征表达。

2023-04-24math.COmath.PRmath.RT

NW-SE expansions of non-symmetric Cauchy kernels on near staircases and growth diagrams

Lascoux has given a triangular version of the Cauchy identity where Schur polynomials are replaced by Demazure characters and Demazure atoms. He has then used the staircase expansion to recover expansions for all Ferrers shapes, where the Demazure characters and Demazure atoms are under the action of Demazure operators specified by the cells above the staircase. The characterisation of the tableau-pairs in these last expansions is less explicit. We give here a bijective proof for expansions over near staircases, where the tableau-pairs are made explicit. Our analysis formulates Mason's RSK ana…

2014-12-01math.COmath.RT

Lakshmibai-Seshadri paths and non-symmetric Cauchy identity

We give a simple crystal theoretic interpretation of the Lascoux's expansion of a non-symmetric Cauchy kernel $\prod_{i+ j\leq n+1}(1-x_iy_j)^{-1}$, which is given in terms of Demazure characters and atoms. We give a bijective proof of the non-symmetric Cauchy identity using the crystal of Lakshmibai-Seshadri paths, and extend it to the case of continuous crystals.

2017-05-18math.COmath.RT

Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product

This paper introduces noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand, one can derive a noncommutative Cauchy identity as well as study a noncommutative scalar product implied by Cauchy identity. This scalar product seems be the noncommutative analog of Hall scalar product in the commutative theory.

2007-12-13math.COmath.RA

An analogue of the Robinson-Schensted-Knuth correspondence and non-symmetric Cauchy kernels for truncated staircases

We prove a restriction of an analogue of the Robinson--Schensted--Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangements of each other, satisfies an inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is bounded by the reverse of the other. For semi-standard Young tableaux the inequality means…

2014-11-26math.COmath.RT

更多相关论文收起更多论文

Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality

This paper studies variance functions of Cauchy-Stieltjes Kernel families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known ones. We construct a class of examples which exhausts all cubic variance functions, and provide examples of polynomial variance functions of arbitrary degree. We also relate Cauchy-Stieltjes Kernel families with polynomial variance functions to generalized orthogonality. Our main results are stated solely in terms of classical probability; some proofs rely on a…

2019-10-12math.COmath.PRmath.ST

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…

2010-10-23math.CO

Refined Cauchy identity for spin Hall-Littlewood symmetric rational functions

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…

2021-07-22math-phmath.COmath.MP

Interpolation Macdonald polynomials and Cauchy-type identities

Let Sym denote the algebra of symmetric functions and $P_μ(\,\cdot\,;q,t)$ and $Q_μ(\,\cdot\,;q,t)$ be the Macdonald symmetric functions (recall that they differ by scalar factors only). The $(q,t)$-Cauchy identity $$ \sum_μP_μ(x_1,x_2,\dots;q,t)Q_μ(y_1,y_2,\dots;q,t)=\prod_{i,j=1}^\infty\frac{(x_iy_jt;q)_\infty}{(x_iy_j;q)_\infty} $$ expresses the fact that the $P_μ(\,\cdot\,;q,t)$'s form an orthogonal basis in Sym with respect to a special scalar product $\langle\,\cdot\,,\,\cdot\,\rangle_{q,t}$. The present paper deals with the inhomogeneous \emph{interpolation} Macdonald symmetric function…

2018-09-26math-phmath.COmath.MP