Relations in the quantum cohomology ring of \(G/B\) (Q1882362)

Let \(G\) be a connected, simply connected, simple, complex Lie group, and \(B\subset G\) a Borel subgroup. The presentation of the quantum cohomology ring of the flag variety \(G/B\) has been first computed by \textit{B.~Kim} [Ann. Math. 149, 129--148 (1999; Zbl 1054.14533)], where he points out its relation with the Toda lattice. In this note, the author gives a shorter proof of this fact. His argument relies on the work of \textit{R.~Goodman} and \textit{N. R.~Wallach} [Commun. Math. Phys. 83, 355--386 (1982; Zbl 0503.22013)] concerning the Toda lattices, and then he applies \textit{B.~Siebert} and \textit{G.~Tian}'s result [Asian J. Math. 1, 679--695 (1997; Zbl 0974.14040)] about the presentation of the quantum cohomology ring of Fano varieties. Using this approach, the author extends in [Adv. Math. 185, 347--369 (2004; Zbl 1137.53348)] the result to the infinite dimensional setting, corresponding to loop groups.