Combinatorial Representation Theory

这个项目是在组合表示理论中，研究在某些线性对称下保持不变的对象。对这些不变(或对称)多项式(以及它们的“共不变”对应项)的研究在数学上有很长的历史。它促进了对代数几何、纽结理论和模理论中对象的计算和组合理解。这个项目试图将这个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认识到。这个项目的组合方面可以被相对较少的数学背景的学生理解和处理，这为本科生和即将到来的研究生打开了研究的可能性。这一提议中的一个中心问题是菲尔兹研究所组合学小组关于对称群的超空间共不变环的结构的一个引人注目的猜想。PI将使用分级对称群模(与Haglund、Shimozono和Wilson合作开发)和标志簇的推广(与Pawlowski合作定义)来研究超空间共不变环。在与威尔逊的合作中，PI正在开发一种适用于超空间商研究的轨道谐和。PI还(与Reineke和Tewari联合工作)通过轨道调和发展对箭图表示理论的Donaldson-Thomas不变量的组合理解。这个项目的最终指导是D‘Addrio，Iraci和Vanden Wyngerd关于四分次对称群模的一个美丽猜想，这个四分次对称群模来自n-空间的两个副本上的微分形式。这一猜想的解决方案将推广海曼关于对角线共不变环的结果。这一奖项反映了NSF的法定使命，并已被认为值得支持，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。

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.