Plane partitions with a “pit”: generating functions and representation theory

Mikhail Bershtein; Boris Feigin; Grigory Merzon

doi:10.1007/s00029-018-0389-z

Plane partitions with a “pit”: generating functions and representation theory

Mikhail Bershtein^*, Boris Feigin, Grigory Merzon

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

38 Scopus citations

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 gl₁ 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 gl_m _| _n. We discuss representation theoretic interpretation of our formulas using q-deformed W-algebra gl_m _| _n.

Original language	English
Pages (from-to)	21-62
Number of pages	42
Journal	Selecta Mathematica, New Series
Volume	24
Issue number	1
DOIs	https://doi.org/10.1007/s00029-018-0389-z
State	Published - 1 Mar 2018
Externally published	Yes

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Publisher Copyright:
© 2018, Springer International Publishing AG, part of Springer Nature.

Keywords

05E10
17B37
20G42
81R10

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10.1007/s00029-018-0389-z

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abstract = "We study plane partitions satisfying condition an + 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 gl1 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 glm | n. We discuss representation theoretic interpretation of our formulas using q-deformed W-algebra glm | n.",

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AU - Feigin, Boris

AU - Merzon, Grigory

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N2 - We study plane partitions satisfying condition an + 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 gl1 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 glm | n. We discuss representation theoretic interpretation of our formulas using q-deformed W-algebra glm | n.

AB - We study plane partitions satisfying condition an + 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 gl1 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 glm | n. We discuss representation theoretic interpretation of our formulas using q-deformed W-algebra glm | n.

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Plane partitions with a “pit”: generating functions and representation theory

Abstract

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Keywords

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