Double Crystal Graphs

Abstract

We show how to expand a non-symmetric Cauchy kernel \( \prod {_{i + _j \leqslant n} (1 - x_i y_j } )^{ - 1} \) in the basis of Demazure characters for type An. The construction involves using the left and right structure of crystal graphs on words, and mostly reduces to properties of thejeu de taquin.We give, without proof, the expansion of a similar kernel in the nilplactic algebra, and define characters in two sets of variables.