Quantum Cohomology of Grassmannians

By Anders Skovsted Buch. Compositio Mathematica, 2001

102 citations recorded by the source corpus.

We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/31e0c6ccad3faabeca9240be3b297e4c28d9ea75

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

A Littlewood–Richardson rule for two-step flag varieties (2009, background): “The small quantum cohomology rings of Grassmannians have been studied extensively (see [3,4] or [5] for references).”
A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties (2016, background): “When X is a Grassmannian of type A, this related space is a two-step partial flag variety of kernel-span pairs [7].”
On the quantum product of Schubert classes (2001, background): “but Giambelli formulas are not yet known in general. Most of what is known about quantum Giambelli formulas comes from computing formulas for degeneracy loci on certain Quot schemes (although A. Buch [11] has recently given a proof for the Grassmannian, and Buch, Kresch, and Tamvakis [12] for some o”
Quantum Pieri rules for isotropic Grassmannians (2008, methodology): “n general position. As in [BKT], we will show that these morphisms are maps to a Lagrangian Grassmannian LG(d,2d) contained in IG whose image curves pass through three general points of LG. Following [Bu], we deﬁne the kernel of a rational map f : P1 → IG as the intersection of all the subspaces Σt=”
Finiteness of cominuscule quantum K-theory (2010, background): “The points of the variety Yd provide a generalization to cominuscule varieties of the kernel-span pairs of curves in classical Grassmannians [3, 4].”