The Quantum Toroidal Algebra $\widehat{\widehat{\mathfrak{gl}_1}}$: Calculation of Characters of Some Representations as Generating Functions of Plane Partitions

The generating function of plane partitions $\{a_{i,j}\}$ subject to the constraint $a_{m,n}=0$ is expressed and calculated as the character of an irreducible representation of the quantum toroidal algebra $\widehat{\widehat{\mathfrak{gl}}}_1$ in the case $K=q_1^mq_2^n$.