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Abstract We give a simple and uniform proof of a conjecture of Haines–Richarz characterizing the smooth locus of Schubert varieties in twisted affine Grassmannians. Our method is elementary and avoids any representation theoretic techniques, instead relying on a combinatorial analysis of tangent spaces of Schubert varieties.
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Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture
Abstract A minimal presentation of the cohomology ring of the flag manifold was given in A. Borel (1953). This presentation was extended by E. Akyildiz–A. Lascoux–P. Pragacz (1992) to a nonminimal one for all Schubert varieties. Work of V. Gasharov–V. Reiner (2002) gave a short, that is, polynomial‐size, presentation for a subclass of Schubert varieties that includes the smooth ones. In V. Reiner–A. Woo–A. Yong (2011), a general shortening was found; it implies an exponential upper bound of on the number of generators required. That work states a minimality conjecture whose significance would be an exponential lower bound of on the number of generators needed in worst case, giving the first obstructions to short presentations. We prove the minimality conjecture. Our proof uses the Hopf algebra structure of the ring of symmetric functions.
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- PAR ID:
- 10482252
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- Volume:
- 109
- Issue:
- 1
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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