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dc.contributor.advisorLandsberg, J. M.
dc.creatorOeding, Luke
dc.date.accessioned2010-07-15T00:12:28Z
dc.date.accessioned2010-07-23T21:43:55Z
dc.date.available2010-07-15T00:12:28Z
dc.date.available2010-07-23T21:43:55Z
dc.date.created2009-05
dc.date.issued2010-07-14
dc.date.submittedMay 2009
dc.identifier.urihttp://hdl.handle.net/1969.1/ETD-TAMU-2009-05-526
dc.description.abstractThe variety of principal minors of nxn symmetric matrices, denoted Zn, can be described naturally as a projection from the Lagrangian Grassmannian. Moreover, Zn is invariant under the action of a group G C GL(2n) isomorphic to (SL(2)xn) x Sn. One may use this symmetry to study the defining ideal of Zn as a G-module via a coupling of classical representation theory and geometry. The need for the equations in the defining ideal comes from applications in matrix theory, probability theory, spectral graph theory and statistical physics. I describe an irreducible G-module of degree 4 polynomials called the hyperdeterminantal module (which is constructed as the span of the G-orbit of Cayley's hyperdeterminant of format 2 x 2 x 2) and show that it that cuts out Zn set theoretically. This result solves the set-theoretic version of a conjecture of Holtz and Sturmfels and gives a collection of necessary and sufficient conditions for when it is possible for a given vector of length 2n to be the principal minors of a symmetric n x n matrix. In addition to solving the Holtz and Sturmfels conjecture, I study Zn as a prototypical G-variety. As a result, I exhibit the use of and further develop techniques from classical representation theory and geometry for studying G-varieties.en
dc.format.mimetypeapplication/pdf
dc.language.isoeng
dc.subjectG-varietiesen
dc.subjectPrincipal minorsen
dc.subjectsymmetric matricesen
dc.subjectinverse eigenvalue problemen
dc.subjectPrincipal minor assignment problemen
dc.subjectRelations among principal minors of symmetric matricesen
dc.titleG-Varieties and the Principal Minors of Symmetric Matricesen
dc.typeBooken
dc.typeThesisen
thesis.degree.departmentMathematicsen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorTexas A&M Universityen
thesis.degree.nameDoctor of Philosophyen
thesis.degree.levelDoctoralen
dc.contributor.committeeMemberLima-Filho, Paulo
dc.contributor.committeeMemberSottile, Frank
dc.contributor.committeeMemberStiller, Peter
dc.contributor.committeeMemberPope, Christopher
dc.type.genreElectronic Dissertationen
dc.type.materialtexten


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