  Bijectivization refines the Yang-Baxter equation into a pair of local Markov
moves which randomly update the configuration of the vertex model. Employing
this approach, we introduce new Yang-Baxter random fields of Young diagrams
based on spin $q$-Whittaker and spin Hall-Littlewood symmetric functions. We
match certain scalar Markovian marginals of these fields with (1) the
stochastic six vertex model; (2) the stochastic higher spin six vertex model;
and (3) a new vertex model with pushing which generalizes the $q$-Hahn
PushTASEP introduced recently by Corwin-Matveev-Petrov (arXiv:1811.06475). Our
matchings include models with two-sided stationary initial data, and we obtain
Fredholm determinantal expressions for the $q$-Laplace transforms of the height
functions of all these models. Moreover, we also discover difference operators
acting diagonally on spin $q$-Whittaker or (stable) spin Hall-Littlewood
symmetric functions.
