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Plane partitions with a “pit”: generating functions and representation theory. (English) Zbl 1430.17044

Summary: 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 \(\mathfrak {gl}_{m|n}\).

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
20G42 Quantum groups (quantized function algebras) and their representations
05E10 Combinatorial aspects of representation theory
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