A Littlewood-Richardson rule for two-step flag varieties. (English) Zbl 1213.14088
The author proves a positive, geometric rule for expressing the structure constants of the cohomology ring of two-step flag varieties in terms of their Schubert basis. A corollary is a positive, geometric rule for expressing the structure constants of the small quantum cohomology ring of Grassmannians. He also shows a similar rule for computing the cohomology class of intersections of projections of Schubert varieties in partial flag manifolds.
The method of the paper is a combinatorial record-keeping of codimension 1 degenerations of subvarieties of Grassmannians and flag varieties. Degeneration methods to study the geometry of these varieties date back to at least Pieri. The challenge of this method is finding natural and canonical degeneration orders and finding the combinatorial objects to keep track of these. The paper under review contains such a choice (a new one), resulting in simple and effective rules. The combinatorial objects the author uses are called Mondrian tableau. They supply a convenient tool for recording the rank data for the intersection of two flags.
The author suggests that the algorithms presented in the paper apply in even more general settings than those shown in the paper.
The method of the paper is a combinatorial record-keeping of codimension 1 degenerations of subvarieties of Grassmannians and flag varieties. Degeneration methods to study the geometry of these varieties date back to at least Pieri. The challenge of this method is finding natural and canonical degeneration orders and finding the combinatorial objects to keep track of these. The paper under review contains such a choice (a new one), resulting in simple and effective rules. The combinatorial objects the author uses are called Mondrian tableau. They supply a convenient tool for recording the rank data for the intersection of two flags.
The author suggests that the algorithms presented in the paper apply in even more general settings than those shown in the paper.
Reviewer: Richárd Rimányi (Chapel Hill)
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 32M10 | Homogeneous complex manifolds |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
Keywords:
Schubert calculus; Littlewood-Richardson rule; two-step flag variety; quantum cohomology; Mondrian tableauReferences:
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