77 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
Even Partitions in Plethysms
We prove that for all natural numbers k,n,d with k <= d and every partition
lambda of size kn with at most k parts there exists an irreducible GL(d,
C)-representation of highest weight 2*lambda in the plethysm Sym^k(Sym^(2n)
(C^d)). This gives an affirmative answer to a conjecture by Weintraub (J.
Algebra, 129 (1):103-114, 1990). Our investigation is motivated by questions of
geometric complexity theory and uses ideas from quantum information theory.Comment: 9 page
Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH
We provide counter-examples to Mulmuley's strong saturation conjecture
(strong SH) for the Kronecker coefficients. This conjecture was proposed in the
setting of Geometric Complexity Theory to show that deciding whether or not a
Kronecker coefficient is zero can be done in polynomial time. We also provide a
short proof of the #P-hardness of computing the Kronecker coefficients. Both
results rely on the connections between the Kronecker coefficients and another
family of structural constants in the representation theory of the symmetric
groups: Murnaghan's reduced Kronecker coefficients.
An appendix by Mulmuley introduces a relaxed form of the saturation
hypothesis SH, still strong enough for the aims of Geometric Complexity Theory.Comment: 25 pages. With an appendix by Ketan Mulmuley. To appear in
Computational Complexity. See also
http://emmanuel.jean.briand.free.fr/publications
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
On the complexity of computing Kronecker coefficients
We study the complexity of computing Kronecker coefficients
. We give explicit bounds in terms of the number of parts
in the partitions, their largest part size and the smallest second
part of the three partitions. When , i.e. one of the partitions
is hook-like, the bounds are linear in , but depend exponentially on
. Moreover, similar bounds hold even when . By a separate
argument, we show that the positivity of Kronecker coefficients can be decided
in time for a bounded number of parts and without
restriction on . Related problems of computing Kronecker coefficients when
one partition is a hook, and computing characters of are also considered.Comment: v3: incorporated referee's comments; accepted to Computational
Complexit
No occurrence obstructions in geometric complexity theory
The permanent versus determinant conjecture is a major problem in complexity
theory that is equivalent to the separation of the complexity classes VP_{ws}
and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a
strengthened version of this conjecture over the complex numbers that amounts
to separating the orbit closures of the determinant and padded permanent
polynomials. In that paper it was also proposed to separate these orbit
closures by exhibiting occurrence obstructions, which are irreducible
representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit
closure, but not in the other. We prove that this approach is impossible.
However, we do not rule out the general approach to the permanent versus
determinant problem via multiplicity obstructions as proposed by Mulmuley and
Sohoni.Comment: Substantial revision. This version contains an overview of the proof
of the main result. Added material on the model of power sums. Theorem 4.14
in the old version, which had a complicated proof, became the easy Theorem
5.4. To appear in the Journal of the AM
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