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Algebraic Metacomplexity and Representation Theory

Authors Maxim van den Berg , Pranjal Dutta , Fulvio Gesmundo , Christian Ikenmeyer , Vladimir Lysikov

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic complexity theory
  • metacomplexity
  • representation theory
  • geometric complexity theory

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Abstract

In the algebraic metacomplexity framework we prove that the decomposition of metapolynomials into their isotypic components can be implemented efficiently, namely with only a quasipolynomial blowup in the circuit size. We use this to resolve an open question posed by Grochow, Kumar, Saks & Saraf (2017). Our result means that many existing algebraic complexity lower bound proofs can be efficiently converted into isotypic lower bound proofs via highest weight metapolynomials, a notion studied in geometric complexity theory. In the context of algebraic natural proofs, it means that without loss of generality algebraic natural proofs can be assumed to be isotypic. Our proof is built on the Poincaré-Birkhoff-Witt theorem for Lie algebras and on Gelfand-Tsetlin theory, for which we give the necessary comprehensive background.

Cite As Get BibTex

Maxim van den Berg, Pranjal Dutta, Fulvio Gesmundo, Christian Ikenmeyer, and Vladimir Lysikov. Algebraic Metacomplexity and Representation Theory. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 26:1-26:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.26

Author Details

Maxim van den Berg
  • Ruhr University Bochum, Germany
  • University of Amsterdam, The Netherlands
Pranjal Dutta
  • Nanyang Technological University, Singapore
Fulvio Gesmundo
  • University of Toulouse, France
Christian Ikenmeyer
  • University of Warwick, UK
Vladimir Lysikov
  • Ruhr University Bochum, Germany

Funding

  • van den Berg, Maxim: supported by the European Research Council through an ERC Starting Grant (grant agreement No. 101040907, SYMOPTIC) and the Dutch National Growth Fund (NGF) as part of the Quantum Delta NL visitor programme.
  • Dutta, Pranjal: The work was done while PD was at the National University of Singapore, funded by NRF Singapore under the project "Computational Hardness of Lattice Problems and Implications" [NRF-NRFI09-0005].
  • Ikenmeyer, Christian: supported by EPSRC grants EP/W014882/1 and EP/W014882/2.
  • Lysikov, Vladimir: supported by the European Research Council through an ERC Starting Grant (grant agreement No. 101040907, SYMOPTIC). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

We are grateful to Alessandro Danelon and Nutan Limaye for helpful discussions. We thank anonymous referees for their comments, and especially for bringing our attention to Open Question 2 in [J. A. Grochow et al., 2017].

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