$q$-Whittaker polynomials: bases, branching and direct limits

We study $q$-Whittaker polynomials and their monomial expansions given by the fermionic formula, the inv statistic of Haglund-Haiman-Loehr and the quinv statistic of Ayyer-Mandelshtam-Martin. The combinatorial models underlying these expansions are partition overlaid patterns and column strict fillings. The former model is closely tied to representations of the affine Lie algebra $\widehat{\mathfrak{sl}_n}$ and admits projections, branching maps and direct limits that mirror these structures in the Chari-Loktev basis of local Weyl modules. We formulate novel versions of these notions in the column strict fillings model and establish their main properties. We construct weight-preserving bijections between the models which are compatible with projection, branching and direct limits. We also establish connections to the coloured lattice paths formalism for $q$-Whittaker polynomials due to Wheeler and collaborators.