Combinatorics and Representation Theory of Rank Varieties, Springer Fibers, and Hyperplane Arrangements

Abstract

This thesis is dedicated to applications of symmetric function theory to problems in combinatorics, representation theory, and geometry. Crucial to our applications is the Frobenius characteristic map from Algebraic Combinatorics, which associates a symmetric function to each finite-dimensional symmetric group module. First, we introduce a family of quotient rings $R_{n,\lambda, s}$ that have the structure of graded symmetric group modules. This family of rings simultaneously generalizes the cohomology rings of Springer fibers studied by Garsia and Procesi and the generalized coinvariant rings of Haglund, Rhoades, and Shimozono. We use techniques developed by Garsia and Procesi to prove formulas for the graded Frobenius characteristic of $R_{n,\lambda, s}$, generalizing previous formulas for Springer fibers and generalized coinvariant rings. We then apply our results to Eisenbud-Saltman rank varieties. Second, we present joint work with Gessel and Tewari in which we prove conjectures of Gessel relating a multivariate generating function $G$ encoding labeled binary trees to symmetric group representations. We prove these conjectures by expanding $G$ positively in terms of ribbon Schur symmetric functions. We then connect specializations of $G$ to symmetric group actions on hyperplane arrangements.