Selecta Mathematica

, Volume 24, Issue 1, pp 21–62 | Cite as

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

  • Mikhail Bershtein
  • Boris Feigin
  • Grigory Merzon


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

Mathematics Subject Classification

17B37 81R10 20G42 05E10 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Mikhail Bershtein
    • 1
    • 2
    • 3
    • 4
    • 5
  • Boris Feigin
    • 3
  • Grigory Merzon
    • 6
  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Skolkovo Institute of Science and TechnologyMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.Institute for Information Transmission ProblemsMoscowRussia
  5. 5.Independent University of MoscowMoscowRussia
  6. 6.Moscow Center for Continuous Mathematical EducationMoscowRussia

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