Algebraic metacomplexity and representation theory

TL;DR

This paper introduces an efficient method for decomposing metapolynomials into isotypic components within the algebraic metacomplexity framework, enabling significant advancements in algebraic complexity lower bounds and natural proofs.

Contribution

It provides a quasipolynomial-time algorithm for isotypic decomposition of metapolynomials, resolving an open problem and connecting algebraic complexity lower bounds with geometric complexity theory.

Findings

Efficient quasipolynomial decomposition of metapolynomials.

Conversion of algebraic complexity lower bounds into isotypic lower bounds.

Simplification of algebraic natural proofs to be isotypic.

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.