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

SIGMA 3 (2007), 029, 12 pages      math.CO/0610719
Contribution to the Vadim Kuznetsov Memorial Issue

The 6 Vertex Model and Schubert Polynomials

Alain Lascoux
Université de Marne-La-Vallée, 77454, Marne-La-Vallée, France

Received October 24, 2006; Published online February 23, 2007

We enumerate staircases with fixed left and right columns. These objects correspond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization factor, to some Schubert polynomials.

Key words: alternating sign matrices; Young tableaux; staircases; Schubert polynomials; integrable systems.

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  1. Bressoud D., Proofs and confirmations: the story of alternating sign matrix conjecture, Cambridge University Press, 1999.
  2. Fomin S., Kirillov A., The Yang-Baxter equation, symmetric functions and Schubert polynomials, Discrete Math. 153 (1996), 123-143.
  3. Gaudin M., La fonction d'onde de Bethe, Masson, 1983.
  4. Hamel A.M., King R.C., Symplectic shifted tableaux and deformations of Weyl's denominator formula for sp(2n), J. Algebraic Combin. 16 (2002), 269-300.
  5. Hamel A.M., King R.C., U-turn alternating sign matrices, symplectic shifted tableaux and their weighted enumeration, math.CO/0312169.
  6. Izergin A.G., Partition function of the six-vertex model in a finite volume, Soviet Phys. Dokl. 32 (1987), 878-879.
  7. Kirillov A., Smirnov F.A., Solutions of some combinatorial problems connected with the computation of correlators in the exact solvable models, Zap. Nauch Sem. Lomi 164 (1987), 67-79 (English transl.: J. Soviet. Mat. 47 (1989), 2413-2422).
  8. Kuperberg G., Another proof of the alternating sign matrix conjecture, Int. Math. Res. Not. 1996 (1996), 139-150, math.CO/9712207.
  9. Lascoux A., Square Ice enumeration, Sém. Lothar. Combin. 42 (1999), Art. B42p, 15 pages.
  10. Lascoux A., Chern and Yang through Ice, Preprint, 2002.
  11. Lascoux A., Symmetric functions & combinatorial operators on polynomials, CBMS Regional Conference Series in Mathematics, Vol. 99, American Mathematical Society, Providence, RI, 2003.
  12. Lascoux A., Schützenberger M.P., Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), no. 2, R27, 35 pages.
  13. Macdonald I.G., Notes on Schubert polynomials, LACIM, Publi. Université Montréal, 1991.
  14. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, Oxford, 1995.
  15. Mills W.H., Robbins D.P., Rumse H.Y., Alternating-sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359.
  16. Okada S., Alternating sign matrices and some deformations of Weyl's denominator formula, J. Algebraic Combin. 2 (1993), 155-176.
  17. Robbins D.P., Rumsey H., Determinants and alternating sign matrices, Adv. Math. 62 (1986), 169-184.
  18. Zeilberger D., Proof of the alternating sign matrix conjecture, Electron. J. Combin. 3 (1996), R13, no. 2, 84 pages, math.CO/9407211.

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