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Two problems in representation theory: affine Lie algebras and algebraic combinatorics
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Two problems in representation theory: affine Lie algebras and algebraic combinatorics
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Ginory
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Alejandro
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1983-
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Alejandro Ginory
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author
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Sahi
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Siddhartha
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Siddhartha Sahi
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chair
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Lepowsky
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James
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James Lepowsky
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internal member
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HUANG
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YI-ZHI
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YI-ZHI HUANG
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internal member
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Hong
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Jiuzu
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Jiuzu Hong
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Advisory Committee
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Rutgers University
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School of Graduate Studies
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theses
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2019
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2019-05
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English
Abstract
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In this dissertation, we investigate two topics with roots in representation theory. The first topic is about twisted affine Kac-Moody algebras and vector spaces spanned bytheir characters. Specifically, the space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of SL (2, Z). By embedding the characters in the space spanned by theta functions, we study an SL (2, Z)-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram automorphisms on conformal blocks, have negative structure constants with respect to the (usual) basis indexed by the dominant integral weights of the Lie algebra. We give positivity conjectures for the new structure constants and prove them in some illuminating cases. We then compute formulas for the action of congruence subgroups on these character spaces and give explicit descriptions of the quotients using the affine Weyl group.
The second topic concerns algebraic combinatorics and symmetric functions. In statistics, zonal polynomials and Schur functions appear when taking integrals over certain compact Lie groups with respect to their associated Haar measures. Recently, a conjecture, related to certain integrals of statistical interest, was proposed by D. Richards and S. Sahi. This conjecture asserts that certain linear combinations of Jack polynomials, a one-parameter family of symmetric polynomials that generalizes the zonal and Schur polynomials, are non-negative when evaluated over a certain cone. In the second part of this dissertation, we investigate these conjectures for Schur polynomials and give a refined version of the conjecture. In addition, we prove some cases and arrive at certain seemingly new combinatorial results. In an important instance, we give an analogous result for Jack polynomials.
The second topic concerns algebraic combinatorics and symmetric functions. In statistics, zonal polynomials and Schur functions appear when taking integrals over certain compact Lie groups with respect to their associated Haar measures. Recently, a conjecture, related to certain integrals of statistical interest, was proposed by D. Richards and S. Sahi. This conjecture asserts that certain linear combinations of Jack polynomials, a one-parameter family of symmetric polynomials that generalizes the zonal and Schur polynomials, are non-negative when evaluated over a certain cone. In the second part of this dissertation, we investigate these conjectures for Schur polynomials and give a refined version of the conjecture. In addition, we prove some cases and arrive at certain seemingly new combinatorial results. In an important instance, we give an analogous result for Jack polynomials.
Subject
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Topic
Affine Lie algebras
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Topic
Mathematics
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Topic
Lie algebras
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Rutgers University Electronic Theses and Dissertations
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ETD_9790
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1 online resource (vii, 90 pages)
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Ph.D.
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Includes bibliographical references
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School of Graduate Studies Electronic Theses and Dissertations
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rucore10001600001
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doi:10.7282/t3-d7ga-4c38
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ETD doctoral
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The author owns the copyright to this work.
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Name
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Ginory
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Alejandro
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2019-04-11 12:54:47
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Alejandro Ginory
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Rutgers University. School of Graduate Studies
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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