Classical Aspects of Quantum Cohomology of Generalized Flag Varieties

Functorial relationships between $QH* (G/B)$ and $QH* (G/P)$, II

“…It is easy to see the next combinatorial fact (see e.g. Lemma 2.8 and Remark 2.9 of [17]). Lemma 3.21.…”

“…[5], [6] and references therein). The special case of the functorial property [16] when P/B ∼ = P 1 has led to nice applications on the "quantum to classical" principle [17], as further applications of which Leung and the author obtained certain quantum Pieri rules [19] as well as alternative proofs of the main results of [4]. On the other hand, our main result could also be treated as a kind of application of the "quantum to classical" principle.…”

“…The latter claim could be further simplified as gr [r,r] (q λ ′ σ w ′ ) < 0, by noting gr [1,r−1] (q λ ′ σ w ′ ) = 0. For this purpose, we need the next main result of [17], which is in fact an application of [16] in the special case of P/B ∼ = P 1 . For each α ∈ ∆, we define a map sgn α : W → {0, 1} by sgn α (w) := 1 if ℓ(w) − ℓ(ws α ) > 0, and 0 otherwise.…”

“…Although Leung and the author have showed an explicit combinatorial formula for those Gromov-Witten invariants (with sign cancelation involved) in [18], it would exceed the capacity of a computer in some cases if we use the formula directly. Instead, we will apply the "quantum to classical" principle developed in [17].…”

W-translated Schubert divisors and transversal intersections

“…There is another type of identities among (equivariant) genus zero, three-point Gromov-Witten invariants for complete flag variety G/B of general Lie types in [29,16]. The special case of degree zero Gromov-Witten invariants recovers the recurrence formula in [22].…”