We use non-symmetric Cauchy kernel identities to get the law of last passage percolation (LPP) models in terms of Demazure characters (which are non symmetric polynomials) of a gl_n Lie algebra. The construction is based on restrictions of the Robinson-Schensted-Knuth (RSK) correspondence to Young shape non negative integer matrices, compatible with crystal basis theory. When the Young shape is a rectangle, we recover the known LPP law given by the Cauchy kernel identity in terms of Schur polynomials (which are symmetric polynomials), gl_n irreducible characters.
The RSK correspondence is a combinatorial bijection between nonnegative integer matrices and pairs of semi-standard Young tableaux of the same shape, with many interesting properties. For each non negative integer mxn matrix the greatest integer which can be obtained starting at the upper right corner (1,n) and ending at the lower left corner (m,1) with down/left path steps coincides with the common length of a longest row in the tableau pair (P,Q). It is then natural to study two dimension percolation models based on the RSK correspondence where random matrices whose entries follow independent geometric distribution are considered. The link to probability has emerged from the known fact that the length of a longest increasing subsequence (encoded in the common length of a longest row in the corresponding Young tableau pair) of a random permutation behaves statistically like the largest eigenvalue of a random Hermitian matrix.
This is a joint work with Thomas Gobet and Cédric Lecouvey.