Factorial Supersymmetric Skew Schur Functions and Ninth Variation Determinantal Identities

Abstract

The determinantal identities of Hamel and Goulden have recently been shown to apply to a tableau-based ninth variation of skew Schur functions. Here we extend this approach and its results to the analogous tableau-based ninth variation of supersymmetric skew Schur functions. These tableaux are built on entries taken from an alphabet of unprimed and primed numbers and that may be ordered in a myriad of different ways, each leading to a determinantal identity. At the level of the ninth variation the corresponding determinantal identities are all distinct but the original notion of supersymmetry is lost. It is shown that this can be remedied at the level of the sixth variation involving a doubly infinite sequence of factorial parameters. Moreover, it is shown that the resulting factorial supersymmetric skew Schur functions are independent of the ordering of the unprimed and primed entries in the alphabet.

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References

  1. 1.

    H. Bachmann, Interpolated Schur multiple zeta values, J. Aust. Math. Soc., 104 (2018), 289–307.

    MathSciNet  Article  Google Scholar 

  2. 2.

    H. Bachmann, S. Charleton, Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta values, European J. Combinatorics 87 (2020), online first.

  3. 3.

    E.A.Bender and D.E. Knuth, Enumeration of plane partitions, J. Combin. Theory A, 13 (1972), 40–54.

    MathSciNet  Article  Google Scholar 

  4. 4.

    A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), 118–175.

    Article  Google Scholar 

  5. 5.

    W.Y.C. Chen, G.-G. Yan, A.L.B. Yang, Transformations of border strips and Schur function determinants, J. Algebraic Combinatorics, 21 (2005) 379–394.

    MathSciNet  Article  Google Scholar 

  6. 6.

    A.M. Foley and R.C. King, Factorial Q-functions and Tokuyama identities for classical Lie groups, Eur. J Combin. 73 (2018), 89–113.

    MathSciNet  Article  Google Scholar 

  7. 7.

    A.M. Foley and R.C. King, Determinantal and Pfaffian identities for ninth variation skew Schur functions and \(Q\)-functions, arXiv: 2002.11796, 26 February 2020.

  8. 8.

    I. Gessel and G. X. Viennot, Determinants, paths, and plane partitions, Preprint, (1989), available at http://people.brandeis.edu/~gessel/homepage/papers/pp.pdf.

  9. 9.

    I.P. Goulden and C. Greene, A new tableau representation for supersymmetric Schur functions, J. Algebra 170 (1994), 687–703.

    MathSciNet  Article  Google Scholar 

  10. 10.

    A.M. Hamel, Pfaffians and determinants for Schur \(Q\)-functions, J. Combinatorial Theory A, 75 (1996), 328–340.

    MathSciNet  Article  Google Scholar 

  11. 11.

    A.M. Hamel and I.P. Goulden, Planar decompositions of tableaux and Schur function determinants, Europ. J. Combinatorics, 16 (1995), 461–477.

    MathSciNet  Article  Google Scholar 

  12. 12.

    R.C. King and S.P.O. Plunkett, The evaluation of weight multiplicities using characters and S-functions, J. Phys. A: Math. Gen. 9 (1976), 863–887.

    MathSciNet  Article  Google Scholar 

  13. 13.

    J.-H. Kwon, A combinatorial proof of a Weyl type formula for hook Schur polynomials, J. Algebraic Combin., 28 (2008), 439–459.

    MathSciNet  Article  Google Scholar 

  14. 14.

    B. Lindström, On the vector representation of induced matroids, Bull. London Math. Soc. 5 (1973), 85–90.

    MathSciNet  Article  Google Scholar 

  15. 15.

    I.G. Macdonald, Schur functions: theme and variations, in Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), Publ. Inst. Rech. Math. Av., Univ. Louis Pasteur, Strasbourg, 498 (1992), 5–39.

  16. 16.

    I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd. Ed., Clarendon Press, Oxford, (1995).

    MATH  Google Scholar 

  17. 17.

    S. Mason, E. Niese, Quasisymmetric (\(k,\ell \))-hook Schur functions, Annals of Combin. 22 (2018), 167–199.

    MathSciNet  Article  Google Scholar 

  18. 18.

    A. Molev, Factorial supersymmetric Schur functions and super Capelli identities, Amer. Math. Soc. Transl. Ser. 2 181, Amer. Math. Soc., Providence, RI, 1998, 109–137.

  19. 19.

    P. Pragacz, Algebro-geometric applications of Schur S- and Q-polynomials, Topics in Invariant Theory, 1991.

  20. 20.

    J.B. Remmel, The combinatorics of \((k,\ell )\)-hook Schur functions, Contemp. Math., 34 (1984), 253–287.

    MathSciNet  Article  Google Scholar 

  21. 21.

    J.B. Remmel, A bijective proof of a factorization theorem for \((k,\ell )\)-hook Schur functions, Lin, and Multilin. Algebra, 28 (1990), 119–154.

    MathSciNet  Google Scholar 

  22. 22.

    M.P. Schützenberger, La correspondence de Robinson, in Combinatoire et Représentationdu Groupes Symétrique D. Foata ed. Lect. Notes in Math. 579 (1977), 59–135.

  23. 23.

    J.R. Stembridge, A characterization of supersymmetric polynomials, J. Algebra, 95 (1985), 439–444.

    MathSciNet  Article  Google Scholar 

  24. 24.

    J.R. Stembridge, Nonintersecting paths, pfaffians and plane partitions. Adv. in Math., 83 (1990), 96–131.

    MathSciNet  Article  Google Scholar 

  25. 25.

    M. Yang, J.B. Remmel, Hook-Schur function analogues of Littlewood’s identities and their bijective proofs, European J. Combin. 19 (1998), 257–272.

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The first author (AMF) was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). The second author (RCK) is grateful for the hospitality extended to him by Professor Bill Chen at the Center for Applied Mathematics at Tianjin University and for the opportunity to pursue this project while visiting him there. This work was supported by the Canadian Tri-Council Research Support Fund.

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Correspondence to Angèle M. Foley.

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Foley, A.M., King, R.C. Factorial Supersymmetric Skew Schur Functions and Ninth Variation Determinantal Identities. Ann. Comb. 25, 229–253 (2021). https://doi.org/10.1007/s00026-021-00526-7

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