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On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions

Authors Julian Dörfler, Christian Ikenmeyer, Greta Panova

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Author Details

Julian Dörfler
  • Saarland University, Saarbrücken, Germany
Christian Ikenmeyer
  • Max Planck Institute for Software Systems, Saarbrücken, Germany
Greta Panova
  • University of Southern Californa, Los Angeles, CA, USA
  • University of Pennsylvania, Philadelphia, PA, USA

Funding

  • Dörfler, Julian: Partially supported by DFG grant IK 116/2-1.
  • Ikenmeyer, Christian: Partially supported by DFG grant IK 116/2-1.
  • Panova, Greta: Partially funded by the NSF.

Acknowledgements

This work was done in part while CI and GP were visiting the Simons Institute for the Theory of Computing.

Cite AsGet BibTex

Julian Dörfler, Christian Ikenmeyer, and Greta Panova. On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.51

Abstract

Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. We provide the first toy setting in which a separation can be achieved for a family of polynomials via these multiplicities. Mulmuley and Sohoni’s papers also conjecture that the vanishing behavior of multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and Bürgisser-Ikenmeyer-Panova (J. AMS). This raises the question whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. We provide first finite settings where a separation via multiplicities can be achieved, while the separation via occurrences is provably impossible. These settings are surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e. a higher secant variety of the Veronese variety). As a side result we prove a slight generalization of Hermite’s reciprocity theorem, which proves Foulkes' conjecture for a new infinite family of cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic complexity theory
  • geometric complexity theory
  • Waring rank
  • plethysm coefficients
  • occurrence obstructions
  • multiplicity obstructions

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References

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