Categorical operators and crystal structures on the ring of symmetric functions ...

• Main Author: Sandoval Gonza?lez, Nicolle Esther author
• Format: Thesis
• Language: English
• Published: University of Southern California Digital Library (USC.DL) 2019
• Subjects: Mathematics degree program; Doctor of Philosophy degree; College of Letters, Arts and Sciences school; sami
• Online Access: https://dx.doi.org/10.25549/usctheses-c89-137584; https://digitallibrary.usc.edu/asset-management/2A3BF1PGYYLL

In this dissertation I prove various results that encompass multiple fields. Within higher representation theory, I categorify the Boson-Fermion correspondence, settling a standing conjecture of Cautis and Sussan. I categorify the creation and annihilation operators for Schur functions, known as Bernstein operators. I also expand the diagrammatic calculus of Khovanov's Heisenberg category by constructing new explicit branching isomorphisms. Moreover, I show that certain categorical vertex operators are Fock space idempotents, proving another series of conjectures of Cautis and Sussan. Within algebraic combinatorics in joint work with Sami Assaf, I enhance the known tools for Demazure crystals by constructing a new axiomatic local characterization for these crystals. We also provide an explicit decomposition of the nonsymmetric Macdonald polynomials as the graded character of Demazure crystals, increasing the known representation theoretic meaning of these polynomials. We then pass to the symmetric setting ...