Combinatorics and Geometry of Symmetric Group Representations

Algebraic combinatorics is an area of mathematics that uses finite and discrete structures to study more complex algebraic and geometric structures. Ideas and techniques from combinatorics are increasingly being used in other areas of pure mathematics such as algebraic geometry and representation theory, as well as applied areas such as mathematical physics and complexity theory.

The central theme of this project is to adapt definitions of classical types of objects in algebraic combinatorics, usually defined perhaps only for partitions or the complete graph, to apply to general diagrams or graphs. Often studying this generalized setting reveals geometric and algebraic connections of which only a shadow is visible in the original setting.

This project includes several directions for research. One direction of study is to investigate the structural relationships between Specht modules for general diagrams, the cohomology classes of diagram Schubert varieties, and the geometry of matching ensemble polytopes. Another topic is to investigate certain weighted lattice point sums on flow polytopes of graphs, which in the case of the complete graph are related to the combinatorics of parking functions and the Hilbert series of the space of diagonal harmonics.

A third topic is to study various generalizations of Schubert polynomials from a geometric and representation-theoretic perspective.