A partial Horn recursion in the cohomology of flag varieties
Edward Richmond
DOI: 10.1007/s10801-008-0149-9
Abstract
Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller” flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain deformation of the cup product defined by Belkale and Kumar (Invent. Math. 166:185-228, 2006). We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written in terms of structure constants coming from induced Grassmannians.
Pages: 1–17
Keywords: keywords partial flag varieties; Horn recursion; Schubert varieties; Schubert calculus; Littlewood-Richardson coefficients; structure constants
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References
1. Belkale, P.: Geometric proofs of Horn and saturation conjectures. J. Algebraic Geom. 15(1), 133-173 (2006)
2. Belkale, P., Kumar, S.: Eigenvalue problem and a new product in cohomology of flag varieties. Invent. Math. 166, 185-228 (2006)
3. Billey, S., Braden, T.: Lower bounds for Kazhdan-Lusztig polynomials from patterns. Trans. Groups 8(4), 321-332 (2003)
4. Fulton, W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.) 37(3), 209-249 (2000) (electronic)
5. Fulton, W.: Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients.
2. Belkale, P., Kumar, S.: Eigenvalue problem and a new product in cohomology of flag varieties. Invent. Math. 166, 185-228 (2006)
3. Billey, S., Braden, T.: Lower bounds for Kazhdan-Lusztig polynomials from patterns. Trans. Groups 8(4), 321-332 (2003)
4. Fulton, W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.) 37(3), 209-249 (2000) (electronic)
5. Fulton, W.: Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients.