eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-07-08
29:1
29:36
10.4230/LIPIcs.CCC.2021.29
article
On the Complexity of Evaluating Highest Weight Vectors
Bläser, Markus
1
Dörfler, Julian
2
https://orcid.org/0000-0002-0943-8282
Ikenmeyer, Christian
3
Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Saarbrücken Graduate School of Computer Science, Saarland Informatics Campus, Germany
University of Liverpool, UK
Geometric complexity theory (GCT) is an approach towards separating algebraic complexity classes through algebraic geometry and representation theory. Originally Mulmuley and Sohoni proposed (SIAM J Comput 2001, 2008) to use occurrence obstructions to prove Valiant’s determinant vs permanent conjecture, but recently Bürgisser, Ikenmeyer, and Panova (Journal of the AMS 2019) proved this impossible. However, fundamental theorems of algebraic geometry and representation theory grant that every lower bound in GCT can be proved by the use of so-called highest weight vectors (HWVs). In the setting of interest in GCT (namely in the setting of polynomials) we prove the NP-hardness of the evaluation of HWVs in general, and we give efficient algorithms if the treewidth of the corresponding Young-tableau is small, where the point of evaluation is concisely encoded as a noncommutative algebraic branching program! In particular, this gives a large new class of separating functions that can be efficiently evaluated at points with low (border) Waring rank. As a structural side result we prove that border Waring rank is bounded from above by the ABP width complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol200-ccc2021/LIPIcs.CCC.2021.29/LIPIcs.CCC.2021.29.pdf
Algebraic complexity theory
geometric complexity theory
algebraic branching program
Waring rank
border Waring rank
representation theory
highest weight vector
treewidth