Nonsymmetric Cauchy kernel, crystals and last passage percolation

We use non-symmetric Cauchy kernel identities to get the law of last passage percolation models in terms of Demazure characters.The construction is based on some restrictions of the RSK correspondence that we rephrase in a unified way which is compatible with crystal basis theory. Contents1 We here consider the paths which are compatible with the version of RSK that will be used in the paper. 2In fact, the convention of our paper differs from that in [26] which considers matrices with nonzero entries in positions (i, j) with 1 ≤ i + j ≤ n + 1 rather than lower-triangular matrices.