Categorical operators and crystal structures on the ring of symmetric functions

Dedication
Acknowledgments
Abstract
Introduction
Part I: Categorical Bernstein operators and the Boson-Fermion correspondence
Chapter A Brief History and Overview
Chapter The Decategorified Story
Fock space
Heisenberg algebra
Clifford algebra
Connections with symmetric functions
The Boson-Fermion correspondence
Chapter Homological Algebra for Infinite Chain Complexes
Chapter Extended Graphical Calculus for Khovanov's Heisenberg Category
Khovanov's Heisenberg category
Diagrammatics for Young idempotents
Littlewood-Richardson branching isomorphisms
Chapter The Categorified Story
Categorical Bernstein operators
A categorical Boson-Fermion correspondence
Chapter Properties of the Categorical Bernstein Operators
Categorifying "4774920 i,j"5775928 =0 and "4774920 *i,*j"5775928 =0.
Proof of Theorem 5.1.3
Categorifying "4774920 i,*j"5775928 =i,j
Proof of Theorem 5.1.4
Chapter Fock Space Idempotents
Part II: Demazure crystals for specialized nonsymmetric Macdonald polynomials
Chapter A Jungle of Symmetric Functions
Nonsymmetric Macdonald polynomials
Demazure modules and Macdonald polynomials
Crystal structures gln and Demazure modules
Main theorems and results
Chapter Macdonald Polynomials
Symmetric polynomials
Nonsymmetric polynomials
Semistandard key tabloids
Chapter Crystals for the General Linear Group
Normal crystals
Demazure modules and their crystals
gln crystals on semistandard Young tableaux
Chapter Characterizations of Demazure Crystals
Extremal subsets of crystals
Local characterizations for Bw()
Demazure lowest weights
Chapter Demazure Crystals on Semistandard Key Tabloids
Crystal operators on key tabloids
Kohnert diagrams and rectification of key tabloids
Demazure property of (C)
Further Examples
Detailed example of the embedding map
Complete example of the Demazure crystals for E(0,3,0,2)(X;q,0)
Chapter Combinatorial Formulas
Hall–Littlewood polynomials
Explicit Demazure expansions
Concluding remarks
Bibliography