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Description: |
The well known Cauchy identity is the best characterization of Schur functions that can be derived using the Robinson-Schensted-Knuth (RSK) correspondence. The Cauchy kernel expands as a sum, over all partitions, of products of Schur functions in variables X and Y separately. The RSK algorithm gives a bijection between words in commutative biletters and pairs of semi-standard Young tableaux of the same shape.
Relaxing the symmetry in the Cauchy kernel, by considering only the boxes in a stair case diagram, this non-symmetric Cauchy kernel is therefore expanded into a subfamily of monomials. From the RSK correspondence we know that this non-symmetric Cauchy kernel can be interpreted as a family of pairs of semi-standard Young tableaux. To characterize them, we use semi-skyline augmented fillings, a combinatorial object due to Haglund, Haiman, and Loehr, which Mason has used to determine the right key of a tableau. There is also an algebraic expansion of the non-symmetric Cauchy kernel in
the basis of Demazure characters for type A by Lascoux. Demazure characters of type A, which are equivalent to key polynomials, have been combinatorial decomposed by Lascoux and Schützenberger. Comparing the combinatorial and the algebraic expansions for the non-symmetric Cauchy kernel, we recover the combinatorial expansion of Demazure characters of type A by Lascoux and Schützenberger. A different bijective proof for the non-symmetric Cauchy identity, based on crystal graphs, has been provided by Lascoux, but we shall not discuss it in this talk.
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