Combinatorial bases in quantum toroidal ${\mathfrak{g}\mathfrak{l}}_2$ modules

We show that many tame modules of the quantum toroidal ${\mathfrak{g}\mathfrak{l}}_2$ algebra can be explicitly constructed in a purely combinatorial way using the theory of $q$-characters. The examples include families of evaluation modules obtained from analytic continuation and automorphism twists of Verma modules of the quantum affine ${\mathfrak{g}\mathfrak{l}}_2$ algebra. The combinatorial bases in the modules are labeled by colored plane partitions with various properties.