Generalized Cauchy identities, trees and multidimensional Brownian motions. Part I: bijective proof of generalized Cauchy identities

A bijective proof of the Cauchy identity for Grothendieck polynomials

We consider pairs of a set-valued column-strict tableau and a reverse plane partition of the same shape. We introduce algortithms for them, which implies a bijective proof for the finite sum Cauchy identity for Grothendieck polynomials and dual Grothendieck polynomials.

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.

Generalized Cauchy identities, trees and multidimensional Brownian motions. Part II: Combinatorial differential calculus

We present an analogue of the differential calculus in which the role of polynomials is played by certain ordered sets and trees. Our combinatorial calculus has all nice features of the usual calculus and has an advantage that the elements of the considered ordered sets might carry some additional information. In this way an analytic proof of generalized Cauchy identities from the previous work of the second author can be directly reformulated in our new language of the combinatorial calculus; furthermore the additional information carried by vertices determines uniquely the bijections present…

Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively

In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström-Gessel-Viennot interpretation of semistandard Young tableaux and the Jacobi-Trudi identity together with elementary observations. After some preparations, this point of view provides very simple "graphical proofs" for classical determinantal identities like the Cauchy--Binet formula, Dodgson's condensation formula, the Plücker relations and Laplace's expansion. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which m…

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.

具有反射对称性的交错符号矩阵与平面分拆：$n+3$ 对等价统计量与一个柯西型恒等式

Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics and a Cauchy-type identity

本文推广了垂直对称交错符号矩阵与带洞六边形环面平铺的等数性，引入 $n+3$ 个参数并证明多变量生成函数一致，还给出辛特征展开，为构造双射和柯西型恒等式提供关键步骤。