Lakshmibai-Seshadri Paths and Non-Symmetric Cauchy Identity

Abstract

We give a simple crystal theoretic interpretation of the Lascoux’s expansion of a non-symmetric Cauchy kernel ${\prod }_{i+ j \leq n + 1}(1-x_{i}y_{j})^{-1}$ , 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.