Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 021, 23 pages      arXiv:1408.2807      https://doi.org/10.3842/SIGMA.2015.021

Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Olivier Blondeau-Fournier and Pierre Mathieu
Département de physique, de génie physique et d'optique, Université Laval, Québec, Canada, G1V 0A6

Received August 28, 2014, in final form February 25, 2015; Published online March 11, 2015

Abstract
Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity.

Key words: symmetric superpolynomials; Schur functions; super tableaux; Pieri rule.

pdf (434 kb)   tex (26 kb)

References

  1. Blondeau-Fournier O., Desrosiers P., Lapointe L., Mathieu P., Macdonald polynomials in superspace as eigenfunctions of commuting operators, J. Comb. 3 (2012), 495-561, arXiv:1202.3922.
  2. Blondeau-Fournier O., Desrosiers P., Lapointe L., Mathieu P., Macdonald polynomials in superspace: conjectural definition and positivity conjectures, Lett. Math. Phys. 101 (2012), 27-47, arXiv:1112.5188.
  3. Blondeau-Fournier O., Desrosiers P., Mathieu P., The supersymmetric Ruijsenaars-Schneider model, arXiv:1403.4667.
  4. Bressoud D.M., Proofs and confirmations. The story of the alternating sign matrix conjecture, MAA Spectrum, Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 1999.
  5. Desrosiers P., Lapointe L., Mathieu P., Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials, Nuclear Phys. B 606 (2001), 547-582, hep-th/0103178.
  6. Desrosiers P., Lapointe L., Mathieu P., Jack polynomials in superspace, Comm. Math. Phys. 242 (2003), 331-360, hep-th/0209074.
  7. Desrosiers P., Lapointe L., Mathieu P., Classical symmetric functions in superspace, J. Algebraic Combin. 24 (2006), 209-238, math.CO/0509408.
  8. Desrosiers P., Lapointe L., Mathieu P., Evaluation and normalization of Jack superpolynomials, Int. Math. Res. Not. 2012 (2012), 5267-5327, arXiv:1104.3260.
  9. Fu A.M., Lascoux A., Non-symmetric Cauchy kernels for the classical groups, J. Combin. Theory Ser. A 116 (2009), 903-917, math.CO/0612828.
  10. Haiman M., Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941-1006, math.AG/0010246.
  11. Ion B., Standard bases for affine parabolic modules and nonsymmetric Macdonald polynomials, J. Algebra 319 (2008), 3480-3517, math.QA/0406060.
  12. Kirillov A.N., Completeness of states of the generalized Heisenberg magnet, J. Sov. Math. 36 (1987), 115-128.
  13. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
  14. Sagan B.E., The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, Vol. 203, 2nd  ed., Springer-Verlag, New York, 2001.

Previous article  Next article   Contents of Volume 11 (2015)