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

SIGMA 3 (2007), 072, 14 pages      arXiv:0705.4546      https://doi.org/10.3842/SIGMA.2007.072
Contribution to the Vadim Kuznetsov Memorial Issue

Skew Divided Difference Operators and Schubert Polynomials

Anatol N. Kirillov
Research Institute of Mathematical Sciences (RIMS), Sakyo-ku, Kyoto 606-8502, Japan

Received May 01, 2007; Published online May 31, 2007

Abstract
We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients.

Key words: divided differences; nilCoxeter algebras; Schubert polynomials.

pdf (252 kb)   ps (187 kb)   tex (15 kb)

References

  1. Bergeron N., Sottile F., Skew Schubert functions and the Pieri formula for flag manifolds, Trans. Amer. Math. Soc. 354 (2002), no. 2, 651-673, alg-geom/9709034.
  2. Chen W., Yan G.-G., Yang A., The skew Schubert polynomials, European J. Combin. 25 (2004), 1181-1196.
  3. Fomin S., Kirillov A.N., The Yang-Baxter equation, symmetric functions and Schubert polynomials, Discrete Math. 53 (1996), 123-143.
  4. Fomin S., Kirillov A.N., Quadratic algebras, Dunkl elements, and Schubert calculus, in Advances in Geometry, Editors J.-L. Brylinski and R. Brylinski, Progr. Math. 172 (1999), 147-182.
  5. Fomin S., Stanley R., Schubert polynomials and nilCoxeter algebra, Adv. Math. 103 (1994), 196-207.
  6. Kirillov A.N., On some quadratic algebras: Jucys-Murphy and Dunkl elements, in Calogero-Moser-Sutherland models (1997, Montrèal, QC), CRM Ser. Math. Phys., Springer, New York, 2000, 231-248, q-alg/9705003.
  7. Kogan M., RC-graphs and a generalized Littlewood-Richardson rule, Int. Math. Res. Not. 15 (2001), 765-782, math.CO/0010108.
  8. Kohnert A., Multiplication of a Schubert polynomial by a Schur polynomial, Ann. Comb. 1 (1997), 367-375.
  9. Lascoux A., Schützenberger M.-P., Polynômes de Schubert, C. R. Acad. Sci. Paris Ser. I Math. 294 (1982), 447-450.
  10. Lenart C., Sottile F., Skew Schubert functions and the Pieri formula for flag manifolds, Trans. Amer. Math. Soc. 354 (2002), 651-673.
  11. Macdonald I.G., Notes on Schubert polynomials, Publ. LACIM, Univ. du Quebec à Montréal, 1991.
  12. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Univ. Press, New York, London, 1995.

Previous article   Next article   Contents of Volume 3 (2007)