eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-07-17
12:1
12:17
10.4230/LIPIcs.CCC.2020.12
article
Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings
Garg, Ankit
1
Ikenmeyer, Christian
2
Makam, Visu
3
Oliveira, Rafael
4
Walter, Michael
5
Wigderson, Avi
6
Microsoft Research, Bangalore, India
University of Liverpool, UK
Institute for Advanced Study, Princeton, NJ, USA
University of Waterloo, Canada
Korteweg-de Vries Institute for Mathematics, Institute for Theoretical Physics, Institute for Logic, Language &Computation, University of Amsterdam, The Netherlands
Institute for Advanced Study, Princeton, NJ, US
We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SL_n(ℂ)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions).
We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol169-ccc2020/LIPIcs.CCC.2020.12/LIPIcs.CCC.2020.12.pdf
generators for invariant rings
succinct encodings