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    G-Varieties and the Principal Minors of Symmetric Matrices

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    OEDING-DISSERTATION.pdf (623.9Kb)
    Date
    2010-07-14
    Author
    Oeding, Luke
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    Abstract
    The 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.
    URI
    http://hdl.handle.net/1969.1/ETD-TAMU-2009-05-526
    Subject
    G-varieties
    Principal minors
    symmetric matrices
    inverse eigenvalue problem
    Principal minor assignment problem
    Relations among principal minors of symmetric matrices
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    • Electronic Theses, Dissertations, and Records of Study (2002– )
    Citation
    Oeding, Luke (2009). G-Varieties and the Principal Minors of Symmetric Matrices. Doctoral dissertation, Texas A&M University. Available electronically from http : / /hdl .handle .net /1969 .1 /ETD -TAMU -2009 -05 -526.

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