论文标题
介绍舒伯特品种的共同点：最小猜想的证明
Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture
论文作者
论文摘要
[A. Borel，1953年]。 [E. Akyildiz-a。 Lascoux-p。 Pragacz，1992年，所有Schubert品种的非最小人物。 [Gasharov-Reiner，2002]的作品提供了简短的，即多项式大小的，呈现了包括光滑的舒伯特品种的子类。在[V。 Reiner-A。 woo-a。 Yong，2011年]，发现了一般的缩短；它意味着所需发电机数量的$ 2^n $的指数上限。这项工作指出了一种最小的猜想，其意义将是$ \ sqrt {2}^{n+2}/\ sqrt {πn} $的指数下限，对最坏情况所需的发电机数量，这给了第一个简短演示的障碍物。我们证明了最小的猜想。我们的证明使用对称函数环的HOPF代数结构。
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.
