1,649 research outputs found
On the C-determinantal range for special classes of matrices
Let A and C be square complex matrices of sizen, the C-determinantal range of A is the subset of the complex plane{det(A−UCU^∗): UU^∗=In}. If A, C are both Hermitian matrices, then by a result of Fiedler (1971)[11] this set is a real line segment.
In our paper we study this set for the case when C is a Hermitian matrix. Our purpose is to revisit and improve two well-known results on this topic. The first result is due to Li concerning theC-numerical range of a Hermitian matrix, see Condition 5.1 (a) in Li, (1994)[20]. The second one is due to C.-K. Li, Y.-T. Poon and N.-S. Sze about necessary and sufficient conditions for the C-determinantal range of A to be a subset of the line, (see Li et al. (2008)[21],
Theorem 3.3)
Topology of Exceptional Orbit Hypersurfaces of Prehomogeneous Spaces
We consider the topology for a class of hypersurfaces with highly nonisolated
singularites which arise as exceptional orbit varieties of a special class of
prehomogeneous vector spaces, which are representations of linear algebraic
groups with open orbits. These hypersurface singularities include both
determinantal hypersurfaces and linear free (and free*) divisors. Although
these hypersurfaces have highly nonisolated singularities, we determine the
topology of their Milnor fibers, complements and links. We do so by using the
action of linear algebraic groups beginning with the complement, instead of
using Morse type arguments on the Milnor fibers. This includes replacing the
local Milnor fiber by a global Milnor fiber which has a complex geometry
resulting from a transitive action of an appropriate algebraic group, yielding
a compact model submanifold for the homotopy type of the Milnor fiber. The
topology includes the (co)homology (in characteristic 0, and 2 torsion in one
family) and homotopy groups, and we deduce the triviality of the monodromy
transformations on rational (or complex) cohomology. The cohomology of the
Milnor fibers and complements are isomorphic as algebras to exterior algebras
or for one family, modules over exterior algebras; and cohomology of the link
is, as a vector space, a truncated and shifted exterior algebra, for which the
cohomology product structure is essentially trivial. We also deduce from Bott's
periodicity theorem, the homotopy groups of the Milnor fibers for determinantal
hypersurfaces in the stable range as the stable homotopy groups of the
associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we
obtain a class of formal linear combinations of exceptional orbit hypersurfaces
which have Milnor fibers which are homotopy equivalent to joins of the compact
model submanifolds.Comment: to appear in the Journal of Topolog
Computing Linear Matrix Representations of Helton-Vinnikov Curves
Helton and Vinnikov showed that every rigidly convex curve in the real plane
bounds a spectrahedron. This leads to the computational problem of explicitly
producing a symmetric (positive definite) linear determinantal representation
for a given curve. We study three approaches to this problem: an algebraic
approach via solving polynomial equations, a geometric approach via contact
curves, and an analytic approach via theta functions. These are explained,
compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in
Systems, Optimization and Control, Birkhauser, Base
Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties
The two-dimensional supersymmetric gauged linear sigma model (GLSM) with
abelian gauge groups and matter fields has provided many insights into string
theory on Calabi--Yau manifolds of a certain type: complete intersections in
toric varieties. In this paper, we consider two GLSM constructions with
nonabelian gauge groups and charged matter whose infrared CFTs correspond to
string propagation on determinantal Calabi-Yau varieties, furnishing another
broad class of Calabi-Yau geometries in addition to complete intersections. We
show that these two models -- which we refer to as the PAX and the PAXY model
-- are dual descriptions of the same low-energy physics. Using GLSM techniques,
we determine the quantum K\"ahler moduli space of these varieties and find no
disagreement with existing results in the literature.Comment: v3: 46 pages, 1 figure. Corrected phase structure of general linear
determinantal varieties. Typos correcte
Phylogenetic Algebraic Geometry
Phylogenetic algebraic geometry is concerned with certain complex projective
algebraic varieties derived from finite trees. Real positive points on these
varieties represent probabilistic models of evolution. For small trees, we
recover classical geometric objects, such as toric and determinantal varieties
and their secant varieties, but larger trees lead to new and largely unexplored
territory. This paper gives a self-contained introduction to this subject and
offers numerous open problems for algebraic geometers.Comment: 15 pages, 7 figure
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