  This work is dedicated to $\mathfrak{sl}_{n+1}$-related integrable stochastic
vertex models; we call such models coloured. We prove several results about
these models, which include the following:
  (1) We construct the basis of (rational) eigenfunctions of the coloured
transfer-matrices as partition functions of our lattice models with certain
boundary conditions. Similarly, we construct a dual basis and prove the
corresponding orthogonality relations and Plancherel formulae; (2) We 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; (3) We show that
our eigenfunctions are certain (non-obvious) reductions of the nested Bethe
Ansatz eigenfunctions; (4) For models in a quadrant with domain-wall (or
half-Bernoulli) boundary conditions, we prove a matching relation that
identifies the distribution of the coloured height function at a point with the
distribution of the height function along a line in an associated colour-blind
($\mathfrak{sl}_2$-related) stochastic vertex model. Thanks to a variety of
known results about asymptotics of height functions of the colour-blind models,
this implies a similar variety of limit theorems for the coloured height
function of our models; (5) We demonstrate how the coloured-uncoloured match
degenerates to the coloured (or multi-species) versions of the ASEP,
$q$-PushTASEP, and the $q$-boson model; (6) We show how our eigenfunctions
relate to non-symmetric Cherednik-Macdonald theory, and we make use of this
connection to prove a probabilistic matching result by applying Cherednik-Dunkl
operators to the corresponding non-symmetric Cauchy identity.
