Geodesic
Quantum Cohomology of Minuscule Homogeneous Spaces III Semi-Simplicity and Consequences
Canadian journal of mathematics, Tome 62 (2010) no. 6, pp. 1246-1263

Voir la notice de l'article provenant de la source Cambridge University Press

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q\,=\,1$ , is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa–Intriligator type formulas for the Gromov–Witten invariants.
DOI : 10.4153/CJM-2010-050-9
Mots-clés : 14M15, 14N35, quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler class 
Chaput, P. E.; Manivel, L.; Perrin, N. Quantum Cohomology of Minuscule Homogeneous Spaces III Semi-Simplicity and Consequences. Canadian journal of mathematics, Tome 62 (2010) no. 6, pp. 1246-1263. doi: 10.4153/CJM-2010-050-9
@article{10_4153_CJM_2010_050_9,
     author = {Chaput, P. E. and Manivel, L. and Perrin, N.},
     title = {Quantum {Cohomology} of {Minuscule} {Homogeneous} {Spaces} {III} {Semi-Simplicity} and {Consequences}},
     journal = {Canadian journal of mathematics},
     pages = {1246--1263},
     year = {2010},
     volume = {62},
     number = {6},
     doi = {10.4153/CJM-2010-050-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-050-9/}
}
TY  - JOUR
AU  - Chaput, P. E.
AU  - Manivel, L.
AU  - Perrin, N.
TI  - Quantum Cohomology of Minuscule Homogeneous Spaces III Semi-Simplicity and Consequences
JO  - Canadian journal of mathematics
PY  - 2010
SP  - 1246
EP  - 1263
VL  - 62
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-050-9/
DO  - 10.4153/CJM-2010-050-9
ID  - 10_4153_CJM_2010_050_9
ER  - 
%0 Journal Article
%A Chaput, P. E.
%A Manivel, L.
%A Perrin, N.
%T Quantum Cohomology of Minuscule Homogeneous Spaces III Semi-Simplicity and Consequences
%J Canadian journal of mathematics
%D 2010
%P 1246-1263
%V 62
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-050-9/
%R 10.4153/CJM-2010-050-9
%F 10_4153_CJM_2010_050_9

[A] [A] Abrams, L., The quantum Euler class and the quantum cohomology of the Grassmannians. Israel J. Math. 117(2000), 335–352. doi:10.1007/BF02773576 Google Scholar

[B] [B] Bertram, A., Quantum Schubert calculus. Adv. Math. 128(1997), no. 2, 289–305. doi:10.1006/aima.1997.1627 Google Scholar

[Bou] [Bou] Bourbaki, N., Groupes et algèbres de Lie. Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968. Google Scholar

[BKT1] [BKT1] Buch, A., Kresch, A., and Tamvakis, H., Gromov–Witten invariants on Grassmannians. J. Amer. Math. Soc. 16(2003), no. 4, 901–915. doi:10.1090/S0894-0347-03-00429-6 Google Scholar

[C MP1] [C MP1] Chaput, P.-E., Manivel, L., and Perrin, N., Quantum cohomology of minuscule homogeneous spaces. Transform. Groups 13(2008), no. 1, 47–89. doi:10.1007/s00031-008-9001-5 Google Scholar

[C MP2] [C MP2] Chaput, P.-E., Manivel, L., Quantum cohomology of minuscule homogeneous spaces II. Hidden symmetries. Int. Math. Res. Not. 2007, no. 22. Google Scholar

[Ch] [Ch] Cheong, D., Quantum cohomology rings of Lagrangian and orthogonal Grassmannians and total positivity. Trans. Amer. Math. Soc. 361(2009), no. 10, 5505–5537. doi:10.1090/S0002-9947-09-04720-5 Google Scholar

[H] [H] Hengelbrock, H., An involution of the quantum cohomology ring of the Grassmannian. . Google Scholar | arXiv

[KT] [KT] Kresch, A. and Tamvakis, H., Quantum cohomology of orthogonal Grassmannians. Compos. Math. 140(2004), no. 2, 482–500. doi:10.1112/S0010437X03000204 Google Scholar

[M] [M] Mehta, L. M., Basic sets of invariant polynomials for finite reflection groups. Comm. Algebra 16(1988), no. 5, 1083–1098. doi:10.1080/00927878808823619 Google Scholar

[P] [P] Postnikov, A., Affine approach to quantum Schubert calculus. Duke Math. J. 128(2005), no. 3, 473–509. doi:10.1215/S0012-7094-04-12832-5 Google Scholar

[R] [R] Rietsch, K., Quantum cohomology rings of Grassmannians and total positivity. Duke Math. J. 110(2001), no. 3, 523–553. doi:10.1215/S0012-7094-01-11033-8 Google Scholar

[ST] [ST] Siebert, B. and Tian, G., On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator. Asian J. Math. 1(1997), no. 4, 679–695. Google Scholar

[T] [T] Tamvakis, H., Quantum cohomology of isotropic Grassmannians. In: Geometric methods in algebra and number theory, Progr. Math., 235, Birkhäuser, Boston, MA, 2005, pp. 311–338. Google Scholar

Cité par Sources :