Plane partitions with a "pit": generating functions and representation theory

TL;DR

This paper derives generating functions for a special class of plane partitions with a 'pit' condition and explores their connections to representation theory of quantum toroidal and Lie superalgebras, providing new character formulas.

Contribution

It introduces explicit formulas for generating functions of plane partitions with a 'pit' and links these to representations of quantum toroidal and Lie superalgebras, offering new insights into their structure.

Findings

Derived explicit generating functions for 'pit' plane partitions.

Connected these formulas to characters of quantum toroidal  algebra representations.

Related formulas to characters of tensor representations of Lie superalgebra _{m|n}.

Abstract

We study plane partitions satisfying condition $a_{n+1,m+1}=0$ (this condition is called "pit") and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra, therefore our formulas can be interpreted as the characters of these representations. The resulting formulas resemble formulas for characters of tensor representations of Lie superalgebra $\mathfrak{gl}_{m|n}$. We discuss representation theoretic interpretation of our formulas using $q$-deformed $W$-algebra $\mathfrak{gl}_{m|n}$.