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Stability of the Heisenberg Product on Symmetric Functions


The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and the Kronecker product. In 1938, Murnaghan discovered that the Kronecker product of two Schur functions stabilizes. We prove an analogous result for the Heisenberg product of Schur functions.

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  1. M. Aguiar, W. Ferrer Santos, and W. Moreira, The Heisenberg product: from Hopf algebras and species to symmetric functions, São Paulo Journal of Mathematical Sciences 11(2) (2017), 261–311.

  2. C. Bowman, M. De Visscher, and R. Orellana, The partition algebra and the Kronecker coefficients, Transactions of the American Mathematical Society 367 (2015), 3647–3667.

    MathSciNet  Article  Google Scholar 

  3. E. Briand, R. Orellana, and M. Rosas, The stability of the Kronecker product of Schur functions, Journal of Algebra 331(1) (2011), 11–27.

    MathSciNet  Article  Google Scholar 

  4. M. Brion, Stable properties of plethysm: on two conjectures of Foulkes, Manuscripta Math 80(1) (1993), 347–371.

    MathSciNet  Article  Google Scholar 

  5. T. Church, J.S. Ellenberg, and B. Farb, Representation stability in cohomology and asymptotics for families of varieties over finite fields, Contemporary Mathematics 620 (2014), 1–54.

    MathSciNet  Article  Google Scholar 

  6. T. Church, J.S. Ellenberg, and B. Farb, FI-module and stability for representtations of symmetric groups, Duke Mathematical Journal 164(9) (2015), 1833–1910.

    MathSciNet  Article  Google Scholar 

  7. T. Church and B. Farb, Representation theory and homological stability, Advances in Mathematics 245 (2013), 250–314.

    MathSciNet  Article  Google Scholar 

  8. I.G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs., The Clarendon Press Oxford University Press, New York, with contributions by A. Zelevinsky, Oxford science publications, 1995.

  9. L. Manivel, On rectangular Kronecker coefficients, J. Algebraic Combin. 33(1) (2011), 153–162.

    MathSciNet  Article  Google Scholar 

  10. Walter Moreira, Products of representations of the symmetric group and non-commutative version, Ph.D. thesis, Texas A &M University, 2008.

  11. F.D. Murnaghan, The analysis of the Kronecker product of irreducible representations of the symmetric group, Amer. J. Math. 60(3) (1938), 761–784.

    MathSciNet  Article  Google Scholar 

  12. B. Sagan, The symmetric group: Representations, Combinatorial Algorithms, and Symmetric Functions, second ed., Graduate Texas in Mathematics., Springer-Verlag, New York, 2001.

  13. Jean-Yves Thibon, Hopf algebras of symmetric functions and tensor products of symmetric group representations, International Journal of Algebra and Computation 1(2) (1991), 207–221.

    MathSciNet  Article  Google Scholar 

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Correspondence to Li Ying.

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Communicated by Matjaž Konvalinka.

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Ying, L. Stability of the Heisenberg Product on Symmetric Functions. Ann. Comb. (2022).

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  • Heisenberg product
  • Kronecker product
  • Schur function

Mathematics Subject Classification

  • 05E05
  • 20C30