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Partially-Symmetric Macdonald Polynomials

Nonsymmetric Macdonald polynomials can be symmetrized in all their variables to obtain the (symmetric) Macdonald polynomials. We generalize this process, symmetrizing the nonsymmetric Macdonald polynomials in only the first k out of n variables. The resulting partially-symmetric Macdonald polynomials interpolate between the symmetric and nonsymmetric types. We begin developing theory for these partially-symmetric polynomials, and prove results including their stability, an integral form, and a Pieri-like formula for their multiplication with certain elementary symmetric functions. / Doctor of Philosophy / There are two well-understood types of polynomials known as the nonsymmetric Macdonald polynomials and symmetric Macdonald polynomials. We define a new form of Macdonald polynomials, which we call partially-symmetric, that are somewhere between the symmetric and nonsymmetric versions. We examine properties of these new partially-symmetric polynomials, including what happens when adding additional symmetric variables, how to multiply them by a constant to clear out denominators in their coefficients, and what happens when multiplying them by another symmetric polynomial.

Identiferoai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/109496
Date29 March 2022
CreatorsGoodberry, Benjamin Nathaniel
ContributorsMathematics, Orr, Daniel D., Mihalcea, Constantin Leonardo, Shimozono, Mark M., Loehr, Nicholas A.
PublisherVirginia Tech
Source SetsVirginia Tech Theses and Dissertation
LanguageEnglish
Detected LanguageEnglish
TypeDissertation
FormatETD, application/pdf
RightsIn Copyright, http://rightsstatements.org/vocab/InC/1.0/

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