University of Southern California Dissertations and Theses

Categorical operators and crystal structures on the ring of symmetric functions

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Categorical operators and crystal structures on the ring of symmetric functions

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Abstract (if available)

Abstract 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 and relate our results to Hall-Littlewood polynomials by using this decomposition to find a new formula for the Kostka-Foulkes polynomials in terms of a simple combinatorial statistic, the major index, which is much easier to compute than previous formulations depending on the more complicated charge statistic.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Sandoval González, Nicolle Esther (author)

Core Title Categorical operators and crystal structures on the ring of symmetric functions

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 04/10/2019

Defense Date 02/28/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag algebraic combinatorics,Bernstein operators,Boson Fermion correspondence,bosons,categorification,CFT,Clifford algebra,combinatorics,creation operators,Crystals,Demazure crystal,Demazure modules,diagrammatic category,fermions,Fock space,Hall-Littlewood polynomials,Heisenberg algebra,higher representation theory,Kostka-Foulkes,Macdonald polynomials,major index,mathematical physics,mathematics,nonsymmetric,OAI-PMH Harvest,QFT,quantum field theory,representation theory,symmetric group,symmetrizers,young idempotents

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lauda, Aaron (committee chair), Assaf, Sami (committee member), Zanardi, Paolo (committee member)

Creator Email nesandov@usc.edu,nicollitasan@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-137584

Unique identifier UC11675581

Identifier etd-SandovalGo-7185.pdf (filename),usctheses-c89-137584 (legacy record id)

Legacy Identifier etd-SandovalGo-7185.pdf

Dmrecord 137584

Document Type Dissertation

Format application/pdf (imt)

Rights Sandoval González, Nicolle Esther

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA