31 research outputs found
Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."
This is a technical report on the proceedings of the workshop held July 21 to
July 25, 2008 at the American Institute of Mathematics, Palo Alto, California,
organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We
include a list of open problems coming from applications in 4 different areas:
signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and
holographic algorithms, and entanglement and quantum information theory. We
emphasize the interactions between geometry and representation theory and these
applied areas
The ideal of the trifocal variety
Techniques from representation theory, symbolic computational algebra, and
numerical algebraic geometry are used to find the minimal generators of the
ideal of the trifocal variety. An effective test for determining whether a
given tensor is a trifocal tensor is also given
Secant cumulants and toric geometry
We study the secant line variety of the Segre product of projective spaces
using special cumulant coordinates adapted for secant varieties. We show that
the secant variety is covered by open normal toric varieties. We prove that in
cumulant coordinates its ideal is generated by binomial quadrics. We present
new results on the local structure of the secant variety. In particular, we
show that it has rational singularities and we give a description of the
singular locus. We also classify all secant varieties that are Gorenstein.
Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain analogous
results for the tangential variety.Comment: Some improvements to previous results, with other minor changes.
Updated reference
Secant varieties of P^2 x P^n embedded by O(1,2)
We describe the defining ideal of the rth secant variety of P^2 x P^n
embedded by O(1,2), for arbitrary n and r at most 5. We also present the Schur
module decomposition of the space of generators of each such ideal. Our main
results are based on a more general construction for producing explicit matrix
equations that vanish on secant varieties of products of projective spaces.
This extends previous work of Strassen and Ottaviani.Comment: 21 page