|
SIGMA 3 (2007), 029, 12 pages math.CO/0610719
https://doi.org/10.3842/SIGMA.2007.029
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
Abstract
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.
pdf (236 kb)
ps (176 kb)
tex (15 kb)
References
- Bressoud D., Proofs and confirmations: the story of alternating
sign matrix conjecture, Cambridge University Press, 1999.
- Fomin S., Kirillov A.,
The Yang-Baxter equation, symmetric functions and Schubert
polynomials, Discrete Math. 153 (1996), 123-143.
- Gaudin M., La fonction d'onde de Bethe, Masson, 1983.
- 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.
- Hamel A.M., King R.C.,
U-turn alternating sign matrices, symplectic shifted tableaux
and their weighted enumeration,
math.CO/0312169.
- Izergin A.G., Partition function of the six-vertex model in a
finite volume, Soviet Phys. Dokl. 32 (1987),
878-879.
- 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).
- Kuperberg G., Another proof of the alternating sign matrix
conjecture, Int. Math. Res. Not. 1996 (1996),
139-150,
math.CO/9712207.
- Lascoux A., Square Ice enumeration, Sém. Lothar. Combin.
42 (1999), Art. B42p, 15 pages.
- Lascoux A., Chern and Yang through Ice, Preprint, 2002.
- Lascoux A., Symmetric functions & combinatorial operators on polynomials,
CBMS Regional Conference Series in Mathematics, Vol. 99,
American Mathematical Society, Providence, RI,
2003.
- Lascoux A., Schützenberger M.P., Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), no. 2, R27,
35 pages.
- Macdonald I.G., Notes on Schubert polynomials, LACIM, Publi.
Université Montréal, 1991.
- Macdonald I.G., Symmetric functions and Hall polynomials, 2nd
ed., Oxford University Press, Oxford, 1995.
- 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.
- Okada S., Alternating sign matrices and some deformations of
Weyl's denominator
formula, J. Algebraic Combin. 2 (1993), 155-176.
- Robbins D.P., Rumsey H., Determinants and alternating sign matrices, Adv. Math. 62 (1986), 169-184.
- Zeilberger D., Proof of the alternating sign matrix conjecture,
Electron. J. Combin. 3 (1996), R13, no. 2, 84 pages,
math.CO/9407211.
|
|