Combinatorial Representation Theory

组合表示理论

• 批准号: 2246846
• 负责人: Brendon Rhoades
• 金额: $ 21.86万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246846&HistoricalAwards=false
• 关键词: Combinatorial; Representation; Theory

This project is in combinatorial representation theory and studies objects which remain invariant under certain linear symmetries. The study of these invariant (or symmetric) polynomials (and their "co-invariant" counterparts) has a long history in mathematics. It has facilitated a computational and combinatorial understanding of objects in algebraic geometry, knot theory, and module theory. This project seeks to extend this program from polynomials to differential forms, objects which play a key role in multivariable calculus but whose combinatorial significance is only now becoming appreciated. The combinatorial aspects of this project can be understood and worked on by students with relatively little mathematical background, which opens research possibilities for undergraduates and incoming graduate students. A central problem in this proposal is a remarkable conjecture of the Fields Institute Combinatorics Group on the structure of the superspace co-invariant ring of the symmetric group. The PI will use graded symmetric group modules (developed in collaboration with Haglund, Shimozono, and Wilson) and a generalization of the flag variety (defined in collaboration with Pawlowski) to study the superspace co-invariant ring. In joint work with Wilson, the PI is developing a version of orbit harmonics which is adapted to the study of superspace quotients. The PI is also (in joint work with Reineke and Tewari) developing a combinatorial understanding of the Donaldson-Thomas invariants of quiver representation theory via orbit harmonics. An ultimate guiding light of this project is a beautiful conjecture of D'Adderio, Iraci, and Vanden Wyngaerd on a quadruply-graded symmetric group module coming from differential forms on two copies of n-space. A resolution to this conjecture would generalize Haiman's results on the diagonal co-invariant ring.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目属于组合表示理论，研究在某些线性对称性下保持不变的对象。对这些不变（或对称）多项式（及其“协不变”对应项）的研究在数学领域有着悠久的历史。它促进了对代数几何、结理论和模理论中对象的计算和组合理解。该项目旨在将该程序从多项式扩展到微分形式，这些对象在多变量微积分中发挥着关键作用，但其组合意义直到现在才被人们所认识。该项目的组合方面可以由数学背景相对较少的学生理解和研究，这为本科生和即将入学的研究生提供了研究可能性。该提案的一个中心问题是菲尔兹研究所组合学组关于对称群的超空间协不变环结构的一个引人注目的猜想。 PI 将使用分级对称群模块（与 Haglund、Shimozono 和 Wilson 合作开发）和旗形变体的推广（与 Pawlowski 合作定义）来研究超空间协不变环。 PI 与威尔逊合作，正在开发一种适用于超空间商研究的轨道谐波版本。 PI 还（与 Reineke 和 Tewari 合作）通过轨道谐波开发对箭袋表示理论的 Donaldson-Thomas 不变量的组合理解。这个项目的最终指路明灯是 D'Adderio、Iraci 和 Vanden Wyngaerd 对来自 n 空间两个副本上的微分形式的四级对称群模块的美丽猜想。这一猜想的解决将概括海曼在对角协不变环上的结果。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。