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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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No occurrence obstructions in geometric complexity theory
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by Peter Bürgisser, Christian Ikenmeyer and Greta Panova HTML | PDF
J. Amer. Math. Soc. 32 (2019), 163-193 Request permission

Abstract:

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes $\mathrm {VP}_{\mathrm {ws}}$ and $\mathrm {VNP}$. In 2001 Mulmuley and Sohoni suggested studying a strengthened version of this conjecture over 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 $\mathrm {GL}_{n^2}(\mathbb {C})$, 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 in 2001.
References
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Additional Information
  • Peter Bürgisser
  • Affiliation: Technische Universität Berlin, Berlin, Germany
  • MR Author ID: 316251
  • Email: pbuerg@math.tu-berlin.de
  • Christian Ikenmeyer
  • Affiliation: Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
  • MR Author ID: 911976
  • Email: cikenmey@mpi-inf.mpg.de
  • Greta Panova
  • Affiliation: University of Pennsylvania, Philadelphia, Pennsylvania; and Institute for Advanced Study, Princeton, New Jersey
  • MR Author ID: 964307
  • Email: panova@math.upenn.edu.
  • Received by editor(s): March 9, 2017
  • Received by editor(s) in revised form: April 20, 2018
  • Published electronically: October 4, 2018
  • Additional Notes: The first author was partially supported by DFG grant BU 1371/3-2.
    The third author was partially supported by NSF grant DMS-1500834
  • © Copyright 2018 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 32 (2019), 163-193
  • MSC (2010): Primary 68Q17, 05E10, 14L24
  • DOI: https://doi.org/10.1090/jams/908
  • MathSciNet review: 3868002