[Spécialisation de polynômes de Grothendieck]
On démontre une formule pour les polynômes de Schubert et de Grothendieck dans le cas de réarrangements du même ensemble de variables. Cette formule généralise les formules usuelles pour ces polynômes en termes de RC-graphes et donne des démonstrations immédiates de plusieurs propriétés importantes de ces polynômes.
We prove a formula for double Schubert and Grothendieck polynomials, specialized to two re-arrangements of the same set of variables. Our formula generalizes the usual formulas for Schubert and Grothendieck polynomials in terms of RC-graphs, and it gives immediate proofs of many other important properties of these polynomials.
Accepté le :
Publié le :
Anders S. Buch 1 ; Richárd Rimányi 2
@article{CRMATH_2004__339_1_1_0,
author = {Anders S. Buch and Rich\'ard Rim\'anyi},
title = {Specializations of {Grothendieck} polynomials},
journal = {Comptes Rendus. Math\'ematique},
pages = {1--4},
publisher = {Elsevier},
volume = {339},
number = {1},
year = {2004},
doi = {10.1016/j.crma.2004.04.015},
language = {en},
}
Anders S. Buch; Richárd Rimányi. Specializations of Grothendieck polynomials. Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 1-4. doi : 10.1016/j.crma.2004.04.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.015/
[1] RC-graphs and Schubert polynomials, Experiment. Math., Volume 2 (1993) no. 4, pp. 257-269 (MR 95g:05107)
[2] A.S. Buch, Alternating signs of quiver coefficients, Preprint, 2003
[3] A.S. Buch, L. Fehér, R. Rimányi, Positivity of quiver coefficients through Thom polynomials, Preprint, 2003
[4] Schur and Schubert polynomials as Thom polynomials—cohomology of moduli spaces, Cent. European J. Math., Volume 4 (2003), pp. 418-434
[5] Grothendieck polynomials and the Yang–Baxter equation, Proc. Formal Power Series and Alg. Comb. (1994), pp. 183-190
[6] The Yang–Baxter equation, symmetric functions, and Schubert polynomials, (Florence, 1993) (Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics), Volume vol. 153 (1996), pp. 123-143 (MR 998b:05101)
[7] Schubert polynomials and the nil-Coxeter algebra, Adv. Math., Volume 103 (1994) no. 2, pp. 196-207 (MR 95f:05115)
[8] Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., Volume 65 (1992) no. 3, pp. 381-420 (MR 93e:14007)
[9] The cohomology ring of weight varieties and polygon spaces, Adv. Math., Volume 160 (2001), pp. 175-204
[10] A. Knutson, E. Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) (2003) in press
[11] A. Knutson, E. Miller, Subword complexes in Coxeter groups, Adv. Math., (2003) in press
[12] The nil Hecke ring and cohomology of G/P for a Kac–Moody group G, Adv. Math., Volume 62 (1986) no. 3, pp. 187-237 (MR 88b:17025b)
[13] The nil Hecke ring and singularity of Schubert varieties, Invent. Math., Volume 123 (1996) no. 3, pp. 471-506 (MR 97j:14057)
[14] Flag varieties and the Yang–Baxter equation, Lett. Math. Phys., Volume 40 (1997) no. 1, pp. 75-90 (MR 98d:05150)
[15] Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., Volume 295 (1982) no. 11, pp. 629-633 (MR 84b:14030)
[16] Symmetrization operators in polynomial rings, Funktsional. Anal. i Prilozhen., Volume 21 (1987) no. 4, pp. 77-78 (MR 89d:16046)
[17] Décompositions dans l'algèbre des différences divisées, Discrete Math., Volume 99 (1992) no. 1–3, pp. 165-179 (MR 93c:20030)
[18] I.G. Macdonald, Notes on Schubert polynomials, Laboratoire de Combinatoire et d'Informatique Mathématique, Université du Québec à Montréal, 1991
[19] Thom polynomials, symmetries and incidences of singularities, Invent. Math., Volume 143 (2001) no. 3, pp. 499-521 (MR 2001k:58082)
[20] On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., Volume 5 (1984) no. 4, pp. 359-372 (MR 86i:05011)
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