Non-symmetric Cauchy kernels

Olga Azenhas,Aram Emami

doi:10.46298/dmtcs.2359

Abstract

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.

Highlights

Given the general Lie algebra gln(C), and its quantum group Uq(gln), finite-dimensional representations of Uq(gln) are classified by the highest weight
Kashiwara (1991) has associated with λ a crystal graph Bλ, which can be realised as a coloured directed graph whose vertices are all semi-standard Young tableaux (SSYTs) of shape λ in the alphabet [n], and the edges are coloured with a colour i, for each pair of crystal operators fi, ei, such that there exists a coloured i-arrow from the vertex P to P if and only if fi(P ) = P, equivalently, ei(P ) = P, for 1 ≤ i ≤ n − 1
Our main Theorem 4.2 exhibits a bijection between biwords in lexicographic order, whose biletters are cell-coordinates in a NW-SE diagonal of a rectangle and below it, containing the biggest stair shape, and pairs of semi-skyline augmented fillings (SSAF) whose shapes satisfy an inequality in the Bruhat order

Summary

Introduction

Given the general Lie algebra gln(C), and its quantum group Uq(gln), finite-dimensional representations of Uq(gln) are classified by the highest weight. Specialising the combinatorial formula for nonsymmetric Macdonald polynomials Eγ(x; q; t), given in Haglund et al (2008), by setting q = t = 0, implies that Eγ(x; 0; 0) is the sum of the monomial weights of all semi-skyline augmented fillings (SSAF) of shape γ which are fillings of composition diagrams with positive integers, weakly decreasing upwards along columns, and the rows satisfy an inversion condition. These polynomials are a decomposition of the Schur polynomial sλ, with γ+ = λ.

Key tableaux a criterion for the Bruhat order in Sn

Definitions and properties

Main Theorem

Expansions of Cauchy kernels in stair-type shapes