Algebraic metacomplexity and representation theory
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 Poincar\'e-Birkhoff-Witt theorem for Lie algebras and on Gelfand-Tsetlin theory, for which we give the necessary comprehensive background.
Cite
@article{arxiv.2411.03444,
title = {Algebraic metacomplexity and representation theory},
author = {Maxim van den Berg and Pranjal Dutta and Fulvio Gesmundo and Christian Ikenmeyer and Vladimir Lysikov},
journal= {arXiv preprint arXiv:2411.03444},
year = {2025}
}
Comments
31 pages. Added Corollary 1.2 adressing an open question from [GKSS]; Introduction is rewritten; some minor fixes