Lakshmibai-Seshadri paths and non-symmetric Cauchy identity

TL;DR

This paper provides a crystal theoretic interpretation of the non-symmetric Cauchy kernel expansion using Lakshmibai-Seshadri paths, offering a bijective proof and extending it to continuous crystals.

Contribution

It introduces a new crystal theoretic perspective on the non-symmetric Cauchy identity through Lakshmibai-Seshadri paths and extends the proof to continuous crystals.

Findings

Crystal interpretation of the non-symmetric Cauchy kernel.

Bijective proof using Lakshmibai-Seshadri paths.

Extension to continuous crystals.

Abstract

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