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.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2009-05-526 |
| Date | 2009 May 1900 |
| Creators | Oeding, Luke |
| Contributors | Landsberg, J. M. |
| Source Sets | Texas A and M University |
| Language | English |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | application/pdf |
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