Algebraic metacomplexity and representation theory

Algebraic metacomplexity and representation theory

We prove that in the algebraic metacomplexity framework, the decomposition of metapolynomials into their isotypic components can be implemented efficiently, namely with only a quasipolynomial blowup in the circuit size. This means that many existing algebraic complexity lower bound proofs can be eff...

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Hauptverfasser: Berg, Maxim van den, Dutta, Pranjal, Gesmundo, Fulvio, Ikenmeyer, Christian, Lysikov, Vladimir
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Computer Science - Computational Complexity
Mathematics - Algebraic Geometry
Mathematics - Representation Theory
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Dutta, Pranjal
Gesmundo, Fulvio
Ikenmeyer, Christian
Lysikov, Vladimir
description We prove that in the algebraic metacomplexity framework, the decomposition of metapolynomials into their isotypic components can be implemented efficiently, namely with only a quasipolynomial blowup in the circuit size. This 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, our result 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.
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subjects Computer Science - Computational Complexity
Mathematics - Algebraic Geometry
Mathematics - Representation Theory
title Algebraic metacomplexity and representation theory
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