Combinatorial Representation Theory

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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.