• Corpus ID: 2439011

# Permanent versus determinant, obstructions, and Kronecker coefficients

@inproceedings{Burgisser2015PermanentVD,
title={Permanent versus determinant, obstructions, and Kronecker coefficients},
author={Peter Burgisser},
year={2015}
}
We give an introduction to some of the recent ideas that go under the name “geometric complexity theory”. We first sketch the proof of the known upper and lower bounds for the determinantal complexity of the permanent. We then introduce the concept of a representation theoretic obstruction, which has close links to algebraic combinatorics, and we explain some of the insights gained so far. In particular, we address very recent insights on the complexity of testing the positivity of Kronecker…
8 Citations

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## References

SHOWING 1-10 OF 75 REFERENCES
Padded Polynomials, Their Cousins, and Geometric Complexity Theory
• Mathematics
ArXiv
• 2012
This work establishes basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and proves asymptotic injectivity of the Foulkes–Howe map.
No Occurrence Obstructions in Geometric Complexity Theory
• Mathematics
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
• 2016
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 and it is proved that the approach to separating these orbit closures by exhibiting occurrence obstructions is impossible.
Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems
• Mathematics
SIAM J. Comput.
• 2001
The notion of a partially stable point in a reductive-group representation is introduced, which generalizes the notion of stability in geometric invariant theory due to Mumford and reduces fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points.
An Overview of Mathematical Issues Arising in the Geometric Complexity Theory Approach to VP≠VNP
• Mathematics
SIAM J. Comput.
• 2011
We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant's algebraic
Explicit lower bounds via geometric complexity theory
• Mathematics
STOC '13
• 2013
We prove the lower bound R Mm) ≥ 3/2 m2-2 on the border rank of m x m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of Mulmuley and
On the Computation of Clebsch–Gordan Coefficients and the Dilation Effect
• Mathematics
Exp. Math.
• 2006
The experiments show that the lattice point algorithm is superior in practice to the standard techniques for computing multiplicities when the weights have large entries but small rank, and provide experimental evidence for two conjectured generalizations of the saturation property of Littlewood–Richardson coefficients.
A quadratic bound for the determinant and permanent problem
• Mathematics
• 2004
The determinantal complexity of a polynomial f is defined here as the minimal size of a matrix M with affine entries such that f = det M. This function gives a minoration of the more traditional size
Geometric Complexity Theory VI : The flip via positivity Dedicated to Sri
An approach based on positivity hypotheses in algebraic geometry and representation theory to implement the flip and thereby resolve the self referential paradox when the underlying field of computation is of characteristic zero.
Characterizing Valiant's Algebraic Complexity Classes
• Mathematics, Computer Science
MFCS
• 2006
Old and new results under a unifying theme are gathered, namely the restrictions imposed upon the gates, building a hierarchy from formulas to circuits, to characterize a uniform version of VNP.