This work is dedicated to \(\mathfrak{sl}_{n+1}\)-related integrable stochastic vertex models; these models are called colored. The authors prove several results about these models, which include the following:

• The authors construct the basis of (rational) eigenfunctions of the colored transfer-matrices as partition functions of their lattice models with certain boundary conditions. Similarly, they construct a dual basis and prove the corresponding orthogonality relations and Plancherel formulae.
• The authors derive a variety of combinatorial properties of those eigenfunctions, such as branching rules, exchange relations under Hecke divided-difference operators, (skew) Cauchy identities of different types, and monomial expansions.
• The authors show that their eigenfunctions are certain (non-obvious) reductions of the nested Bethe Ansatz eigenfunctions.
• For models in a quadrant with domain-wall (or half-Bernoulli) boundary conditions, the authors prove a matching relation that identifies the distribution of the colored height function at a point with the distribution of the height function along a line in an associated color-blind (\(\mathfrak{sl}_{2}\)-related) stochastic vertex model. Thanks to a variety of known results about asymptotics of height functions of the color-blind models, this implies a similar variety of limit theorems for the colored height function of their models.
• The authors demonstrate how the colored/uncolored match degenerates to the colored (or multi-species) versions of the ASEP, \(q\)-PushTASEP, and the \(q\)-boson model.
• The authors show how their eigenfunctions relate to non-symmetric Cherednik-Macdonald theory, and they make use of this connection to prove a probabilistic matching result by applying Cherednik-Dunkl operators to the corresponding non-symmetric Cauchy identity.