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Quantum Cohomology of Minuscule Homogeneous Spaces III Semi-Simplicity and Consequences

Published online by Cambridge University Press:  20 November 2018

P. E. Chaput ,
L. Manivel  and
N. Perrin
P. E. Chaput*
Affiliation:
Laboratoire de Mathématiques Jean Leray, UFR Sciences et Techniques, Nantes, France
L. Manivel*
Affiliation:
Institut Fourier, Université de Grenoble I, Saint-Martin d’Héres, France
N. Perrin*
Affiliation:
Institut de Mathématiques, Université Pierre et Marie Curie, PARIS, France
*
pierre-emmanuel.chaput@math.univ-nantes.fr
Laurent.Manivel@ujf-grenoble.fr
nperrin@math.jussieu.fr
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Abstract

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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.

Keywords

14M1514N35quantum cohomologyminuscule homogeneous spacesSchubert calculusquantum Euler class
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 6 , 14 December 2010 , pp. 1246 - 1263
Copyright
Copyright © Canadian Mathematical Society 2010

References

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