  We review some algebraic and combinatorial structures that underlie models in
the KPZ universality class.Emphasis is placed on the Robinson-Schensted-Knuth
correspondence and its geometric lifting due to A.N.Kirillov. We present how
these combinatorial constructions are used to analyse the structure of solvable
models in the KPZ class and lead to computation of their statistics via
connecting to representation theoretic objects such as Schur, Macdonald and
Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also
present how fundamental representation theoretic concepts, such as the Cauchy
identity, the Pieri rule and the branching rule, can be used, alongside RSK
correspondences, and can be combined with probabilistic ideas, in order to
construct integrable stochastic dynamics on two dimensional arrays of
Gelfand-Tsetlin type, in ways that couple different one dimensional stochastic
processes. For example, interacting particle systems, on the one hand, and
processes related to eigenvalues of random matrices, on the other, thus
illuminating the emergence of random matrix distributions in interacting
stochastic processes. The goal of the notes is to expose some of the
overarching principles, which have driven a significant number of developments
in the field.
