Jacobi-Trudi Type Formula for Character of Irreducible Representations of \(\frak {gl}(m|1)\)

Abstract

We prove a determinantal type formula to compute the irreducible characters of the general Lie superalgebra \(\mathfrak {gl}(m|1)\) in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula was conjectured by J. van der Jeugt and E. Moens for the Lie superalgebra \(\frak {gl}(m|n)\) and generalizes the well-known Jacobi-Trudi formula.

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Acknowledgements

The authors would like to thank VIASM for the financial support and the excellent working environment.

Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant number 101.04-2016.19. A part of this work was carried out when the first and the third named authors were visiting the Vietnam Institute for Advanced Study in Mathematics.

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Correspondence to Phùng Hô Hai.

Additional information

Dedicated to Professor Lê Tuân Hoa on the occasion of his sixtieth birthday

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Bình, N.L.T., Dung, N.T.P. & Hai, P.H. Jacobi-Trudi Type Formula for Character of Irreducible Representations of \(\frak {gl}(m|1)\). Acta Math Vietnam 44, 603–615 (2019). https://doi.org/10.1007/s40306-018-0280-1

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Keywords

  • Character formula
  • Jacobi-trudi type formula
  • Character formula of Lie superalgebra

Mathematics Subject Classification (2010)

  • 17B10
  • 17B15
  • 17B20
  • 17B22