Work

Algebraic metacomplexity and representation theory

Year: 2024

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2411.03444

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

Institutions: University of Amsterdam, Ruhr University Bochum, Centre National de la Recherche Scientifique, Université de Toulouse, Institut de Mathématiques de Toulouse

Language: English

Topic: Complexity and Algorithms in Graphs

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: European Commission, Engineering and Physical Sciences Research Council, HORIZON EUROPE European Research Council

Awards: Algebraic complexity theory via the algebraic geometry and representation theory of generalised continued fractions, Symmetry and Optimization at the Frontiers of Computation

OpenAlex ID: W4404354548

Cites:

Cited by:

Related to: 10