C_2-equivariant Schubert Calculus for homogeneous spaces

齐次空间的 C_2 等变舒伯特微积分

• 批准号: 405468058
• 负责人: Dr. Thomas Hudson
• 金额: --
• 依托单位: Arbeitsgruppe Topologie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/405468058?language=en
• 关键词: C_2; equivariant; Schubert; Calculus; homogeneous

Our goal is to lift classical Schubert calculus result from ordinary non-equivariant cohomology of Grassmannians and flag manifolds to the C_C-equivariant setting. To do this we will use recent developments in equivariant homotopy theory. In particular, the existence of pushforwards along bundle maps as developed by Costenoble and Waner. Specifically, we wish to do this for projective bundles associated to the universal bundles that such Grassmannians and flag manifolds come naturally equipped with. This will lead us to a notion of Segre classes, which will constitute the basic building block of a Giambelli formula expressing the fundamental classes of all Schubert varieties. On the other hand, in the special case of P^1-bundles, this will allow us to define an analogue of divided difference operators on the equivariant cohomology of flag manifolds. We would then like to compute the structure constants of equivariant real and complex Grassmannians. While many computations have been undertaken by Costenoble, Dugger, and Hogle, those computations do not include the data of these structure constants. Further, there is not yet an adequate interpretation of the relevant cohomology in terms of characteristic classes of vector bundles. Using techniques and insight gained from generalized Schubert Calculus, in the setting of algebraic cobordism, we hope to understand the appropriate C_2-equivariant generalization of the relevant combinatorial structure. We also hope to find the appropriate generalization of the ring of symmetric functions where the polynomials representing the Schubert classes live. Using these particular methods will ensure that our results will lend themselves to the analogous computations in the associated equivariant motivic cohomology theories.

我们的目标是将古典舒伯特微积分提升，从普通的格拉曼尼亚人的普通非等级共同体和旗帜歧管引起到C_C-Equivariant环境。为此，我们将使用近代同型理论中的最新发展。特别是，Costenoble和Waner开发的沿捆绑图上的推动力存在。具体来说，我们希望为与普遍的捆绑包相关的投影捆绑包，而这些捆绑包和旗帜歧管自然而然地配备了这种捆绑包。这将导致我们进入Segre类的概念，这将构成Giambelli公式的基本组成部分，表达所有Schubert品种的基本类别。另一方面，在p^1捆绑的特殊情况下，这将使我们能够在旗帜歧管的等模量同居学上定义分开的差异操作员的类似物。然后，我们想计算模棱两可的真实和复杂的司司人的结构常数。尽管Costenoble，Dugger和Hogle进行了许多计算，但这些计算并不包括这些结构常数的数据。此外，还尚无对相关的共同体的充分解释。在代数恢复主义的设置中，利用从广义的舒伯特演算中获得的洞察力，我们希望了解相关组合结构的适当C_2-等级概括。我们还希望找到对称函数环的适当概括，其中代表舒伯特类的多项式现场。使用这些特殊的方法将确保我们的结果将使自己符合相关的模棱两可的共同体学理论中的类似计算。