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

SIGMA 17 (2021), 054, 29 pages      arXiv:2011.05886

Nonsymmetric Macdonald Superpolynomials

Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA

Received November 16, 2020, in final form May 13, 2021; Published online May 23, 2021

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.

Key words: superpolynomials; Hecke algebra; symmetrization; norms.

pdf (549 kb)   tex (38 kb)  


  1. Baker T.H., Forrester P.J., A $q$-analogue of the type $A$ Dunkl operator and integral kernel, Int. Math. Res. Not. 1997 (1997), 667-686, arXiv:q-alg/9701039.
  2. Baker T.H., Forrester P.J., Symmetric Jack polynomials from non-symmetric theory, Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001.
  3. 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.
  4. Braverman A., Etingof P., Finkelberg M., Cyclotomic double affine Hecke algebras (with an appendix by Hiraku Nakajima and Daisuke Yamakawa), Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), 1249-1312, arXiv:1611.10216.
  5. Cherednik I., Nonsymmetric Macdonald polynomials, Int. Math. Res. Not. 1995 (1995), 483-515, arXiv:q-alg/9505029.
  6. Desrosiers P., Lapointe L., Mathieu P., Jack polynomials in superspace, Comm. Math. Phys. 242 (2003), 331-360, arXiv:hep-th/0209074.
  7. Dipper R., James G., Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20-52.
  8. Dunkl C.F., A positive-definite inner product for vector-valued Macdonald polynomials, Sém. Lothar. Combin. 80 (2019), Art. B80a, 26 pages, arXiv:1808.05251.
  9. Dunkl C.F., Luque J.G., Vector valued Macdonald polynomials, Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875.
  10. González C., Lapointe L., The norm and the evaluation of the Macdonald polynomials in superspace, European J. Combin. 83 (2020), 103018, 30 pages, arXiv:1808.04941.
  11. Griffeth S., Orthogonal functions generalizing Jack polynomials, Trans. Amer. Math. Soc. 362 (2010), 6131-6157, arXiv:0707.0251.
  12. Lascoux A., Yang-Baxter graphs, Jack and Macdonald polynomials, Ann. Comb. 5 (2001), 397-424.
  13. Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
  14. Mimachi K., Noumi M., A reproducing kernel for nonsymmetric Macdonald polynomials, Duke Math. J. 91 (1998), 621-634, arXiv:q-alg/9610014.

Previous article  Next article  Contents of Volume 17 (2021)