Symmetries of Gromov-Witten invariants (Q2752199)

The aim of this paper is to present certain symmetric properties of the Gromov-Witten invariants for type A complex flag manifolds. Several related results were found by \textit{N. Bergeron} and \textit{F. Sottile} [Duke Math. J. 95, 373--423 (1998; Zbl 0939.05084)] and \textit{S. Agnihotri} and \textit{C. Woodward} [Math. Res.Lett. 5, No. 6, 817--836 (1998; Zbl 1004.14013)]. Denoting by \(*\) the quantum product of the Schubert classes in the (small) quantum cohomology ring of the flag manifold, the quantum product being defined as \( \sigma_u * \sigma_v = \sum_{ w \in S_n} C_{u,v,w} \sigma_{w_0 w}\) where \(w_0\) is the longest permutation in the symmetric group of \(n\) letters, \(S_n \). After defining precisely the Gromov-Witten invariants the author states five basic properties for them. The author states a formula (proposition 1) for \(C_{u,v,w}\) which he claims follows from propositions 3 and 4 of those five basic ones. Proposition 2 is stated and taken from a previous paper by \textit{S. Fomin, S. Gelfand} and \textit{A. Postnikov} [J. Am. Math. Soc. 10, No. 3, 565--596 (1997; Zbl 0912.14018)]. Let \( o = (1,2, \dots, n) \) be the cyclic permutation of \(S_n\) given by \(o(i) = i + 1 \) for \( i= 1, \ldots, n-1\), \(o(n) = 1\) and \( q_{ij} = q_i q_{i+1} \dots q_{j-1}\) for \( i <j \). Define \( q_{ij} = q_{ji}^{-1}\) for \(i > j \) and \( q_{ii}= 1\). The author proves:NEWLINENEWLINEFor any \( u,v,w \in S_n \), \( C_{u,v,w} = q_{ij} C_{u,o^{-1}v, ow} \) where \( i= v^{-1}(1)\) and \( j = w^{-1}(n) \) (theorem 4). The proof of this theorem is given in section 4.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00024].