Introduction The quantum cohomology ring of a KK ahler manifold X is a deformation of the usual cohomology ring which appears naturally in theoretical physics in the study of the supersymmetric nonlinear sigma models with target X. In W], Witten introduces the quantum multiplication of cohomology classes on X as a certain deformation of the usual… (More)
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.
IONU¸T CIOCAN-FONTANINE 0. Introduction. The main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all projective homogeneous spaces SL n (C)/P , where P is a parabolic subgroup. The quantum cohomology ring of a smooth projective variety, or more generally of a symplectic manifold X, has been… (More)
Based on the ideas in [CKP], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk… (More)
Following the work of Okounkov-Pandharipande [OP1, OP2], Dia-conescu [D], and the recent work [CDKM] studying the equivariant quantum coho-mology QH * (C *) 2 (Hilb n) of the Hilbert scheme and the relative Donaldson-Thomas theory of P 1 × C 2 , we establish a connection between the J-function of the Hilbert scheme and a certain combinatorial identity in… (More)
The famous hook-length formula is a simple consequence of the branching rule for the hook lengths. While the Greene-Nijenhuis-Wilf probabilistic proof is the most famous proof of the rule, it is not completely combinatorial, and a simple bijection was an open problem for a long time. In this extended abstract, we show an elegant bijective argument that… (More)
We propose an approach via Frobenius manifolds to the study (began in [BCK2]) of the relation between rational Gromov–Witten invari-ants of nonabelian quotients X//G and those of the corresponding " abelian-ized " quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of… (More)