Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences

Pierre-Emmanuel Chaput (1) , Laurent Manivel (2) , Nicolas Perrin (3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 IF - Institut Fourier
3 IMJ - Institut de Mathématiques de Jussieu

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 a previous paper. We deduce Vafa-Intriligator type formulas for the Gromov-Witten invariants.

Mots clés: quantum cohomology; minuscule homogeneous spaces; Schubert calculus; quantum Euler class

Domaines: Géométrie algébrique [math.AG]

Pierre-Emmanuel Chaput, Laurent Manivel, Nicolas Perrin. Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences. Canadian Journal of Mathematics, 2010. ⟨hal-00177146⟩