On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions
Abstract
Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. The papers also conjecture that the vanishing behavior of these multiplicities would be sufficient to separate complexity classes (socalled occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, IkenmeyerPanova (Adv. Math.) and BürgisserIkenmeyerPanova (J. AMS). This raises the question whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. This paper provides for the first time a setting where separating with multiplicities can be achieved, while the separation with occurrences is provably impossible. Our setting is surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the socalled Chow variety) and the variety of polynomials of bounded border Waring rank (i.e. a higher secant variety of the Veronese variety). As a side result we prove a slight generalization of Hermite's reciprocity theorem, which proves Foulkes' conjecture for a new infinite family of cases.
 Publication:

arXiv eprints
 Pub Date:
 January 2019
 arXiv:
 arXiv:1901.04576
 Bibcode:
 2019arXiv190104576D
 Keywords:

 Computer Science  Computational Complexity;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 Mathematics  Representation Theory;
 68Q17;
 05E10
 EPrint:
 24 pages