ENGLISH

Categorical operators and crystal structures on the ring of symmetric functions

By Nicolle Esther Sandoval González

Book information

Year
2019
Language
english
Format
PDF
Filesize
2 MB (2211431 bytes)
Series
PhD thesis at University of Southern California
Pages
\213
Topic
Mathematics Algebra
Time added
2022-04-15 04:09:20

Description

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

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