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 ${\prod }_{i+j\le n+1}(1-x_i{y}_j{)}^{-1}$, 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.

16 pages, introduction and abstract revised, minor typos corrected