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

Abstract

We show that many tame modules of the quantum toroidal ${\mathfrak{gl}}_2$ 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 ${\mathfrak{gl}}_2$ algebra. The combinatorial bases in the modules are labeled by colored plane partitions with various properties.

Latex, 30 pages, 12 figures. Misprints corrected, details of the proofs added