2026-07-20T05:06:48Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23720 2025-10-27T10:42:49Z ddc:004 open_access

Algebraic Metacomplexity and Representation Theory van den Berg, Maxim Dutta, Pranjal Gesmundo, Fulvio Ikenmeyer, Christian Lysikov, Vladimir Algebraic complexity theory metacomplexity representation theory geometric complexity theory 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Maxim van den Berg and Pranjal Dutta and Fulvio Gesmundo and Christian Ikenmeyer and Vladimir Lysikov 2025 Is Part Of LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2025.26 urn:nbn:de:0030-drops-237209 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.26 eng https://creativecommons.org/licenses/by/4.0/legalcode