Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929285400363008 |
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| author | Dizier, Avery St. Yong, Alexander |
| author_facet | Dizier, Avery St. Yong, Alexander |
| contents | A minimal presentation of the cohomology ring of the flag manifold $GL_n/B$ was given in [A. Borel, 1953]. This presentation was extended by [E. Akyildiz-A. Lascoux-P. Pragacz, 1992] to a non-minimal one for all Schubert varieties. Work of [Gasharov-Reiner, 2002] gave a short, i.e. 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 $2^n$ on the number of generators required. That work states a minimality conjecture whose significance would be an exponential lower bound of $\sqrt{2}^{n+2}/\sqrt{πn}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_02011 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture Dizier, Avery St. Yong, Alexander Combinatorics Algebraic Geometry 05E14 A minimal presentation of the cohomology ring of the flag manifold $GL_n/B$ was given in [A. Borel, 1953]. This presentation was extended by [E. Akyildiz-A. Lascoux-P. Pragacz, 1992] to a non-minimal one for all Schubert varieties. Work of [Gasharov-Reiner, 2002] gave a short, i.e. 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 $2^n$ on the number of generators required. That work states a minimality conjecture whose significance would be an exponential lower bound of $\sqrt{2}^{n+2}/\sqrt{πn}$ 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. |
| title | Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture |
| topic | Combinatorics Algebraic Geometry 05E14 |
| url | https://arxiv.org/abs/2209.02011 |