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We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263–277). We also give a geometric proof of the quantum Monk formula.
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Ionuţ Ciocan-Fontanine (1999) studied this question.
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