This project aims to study and further develop combinatorial objects appearing in the representation theory of Coxeter groups, Lie algebras or their generalizations (complex reflections groups, Kac-Moody algebras), and simultaneously, to use them for investigating discrete probabilistic models and their connections with problems in mathematical physics. There are numerous interactions between models of these types based on the combinatorics of partitions (conditioning random walks, percolation problems, Tasep, card shuffling, cut-off phenomenon). The aim this project is to develop these interactions by using new results and objects that were introduced recently in representation theory (crystal graphs, shifted Schur functions, Hall-Littlewood and Macdonald polynomials, basic sets, generalizations of the RSK-procedure etc.). One of the original features of this project is to propose a unified approach to these different themes.