Olga Azenhas ; Aram Emami - Non-symmetric Cauchy kernels

dmtcs:2359 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2359
Non-symmetric Cauchy kernelsArticle

Authors: Olga Azenhas 1; Aram Emami 1

  • 1 Center for Mathematics [Coimbra]

Using an analogue of the Robinson-Schensted-Knuth (RSK) algorithm for semi-skyline augmented fillings, due to Sarah Mason, we exhibit expansions of non-symmetric Cauchy kernels $∏_(i,j)∈\eta (1-x_iy_j)^-1$, where the product is over all cell-coordinates $(i,j)$ of the stair-type partition shape $\eta$ , consisting of the cells in a NW-SE diagonal of a rectangle diagram and below it, containing the biggest stair shape. In the spirit of the classical Cauchy kernel expansion for rectangle shapes, this RSK variation provides an interpretation of the kernel for stair-type shapes as a family of pairs of semi-skyline augmented fillings whose key tableaux, determined by their shapes, lead to expansions as a sum of products of two families of key polynomials, the basis of Demazure characters of type A, and the Demazure atoms. A previous expansion of the Cauchy kernel in type A, for the stair shape was given by Alain Lascoux, based on the structure of double crystal graphs, and by Amy M. Fu and Alain Lascoux, relying on Demazure operators, which was also used to recover expansions for Ferrers shapes.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: Non-symmetric Cauchy kernels,Demazure character,key polynomial,Demazure operator,semi-skyline augmented filling,RSK analogue.,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Strategic Project - UI 324 - 2011-2012; Funder: Fundação para a Ciência e a Tecnologia, I.P.; Code: PEst-C/MAT/UI0324/2011

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