[Paper Review] On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions

Julian Dörfler, Christian Ikenmeyer+1 more|arXiv (Cornell University)|Jan 14, 2019

Polynomial and algebraic computation4 citations

TL;DR

This paper establishes the first provably stronger separation using representation-theoretic multiplicity obstructions compared to occurrence obstructions in geometric complexity theory. It demonstrates that for the Chow variety of products of linear forms and the variety of polynomials of bounded border Waring rank, multiplicity obstructions can separate the varieties even when occurrence obstructions fail, providing a concrete setting where multiplicities offer strictly more power than occurrences.

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 (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and B\\"urgisser-Ikenmeyer-Panova (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 so-called 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.

Motivation & Objective

• To resolve Scott Aaronson's open question on whether multiplicity obstructions can be provably stronger than occurrence obstructions in geometric complexity theory.
• To identify a natural algebraic geometry setting where multiplicity obstructions succeed in separating varieties while occurrence obstructions fail.
• To prove a generalized Hermite reciprocity theorem that resolves Foulkes' conjecture for a new infinite family of cases.
• To provide the first explicit example in geometric complexity theory where representation-theoretic multiplicities yield separation not achievable via occurrence obstructions.

Proposed method

• The authors analyze the action of GL_m on the space of homogeneous polynomials of degree n in m variables, focusing on two GL_m-invariant subvarieties: the Chow variety and the higher secant variety of the Veronese variety.
• They compare the representation-theoretic multiplicities of irreducible representations in the coordinate rings of the Chow variety and the Waring rank variety, using character theory and symmetric function techniques.
• The key technical tool is a generalization of Hermite's reciprocity theorem, which allows the authors to compute and compare multiplicities in the coordinate rings of the two varieties.
• They show that while the occurrence obstruction criterion fails (as previously proven in Ikenmeyer-Panova and Bürgisser-Ikenmeyer-Panova), the multiplicity obstruction criterion succeeds due to non-vanishing multiplicities in the coordinate ring of the Chow variety.
• The proof relies on the fact that the coordinate ring of the Chow variety contains certain irreducible representations with non-zero multiplicity, while the Waring rank variety does not, even though both are GL_m-varieties.
• The authors use the structure of symmetric polynomials and plethysm coefficients to establish that the multiplicity obstruction is non-vanishing in the Chow variety setting.

Experimental results

Research questions

• RQ1 Can multiplicity obstructions in geometric complexity theory be strictly stronger than occurrence obstructions in any natural setting?
• RQ2 Is there a setting where the coordinate ring of one GL_m-variety contains irreducible representations with non-zero multiplicity that are absent in another, despite both being defined via symmetric constructions?
• RQ3 Does a generalized Hermite reciprocity theorem imply new cases of Foulkes' conjecture?
• RQ4 Can the failure of occurrence obstructions be overcome by considering multiplicities in the coordinate ring of the Chow variety?
• RQ5 Is there a provably minimal setting where multiplicity obstructions succeed where occurrence obstructions fail?

Key findings

• The paper constructs the first known example where multiplicity obstructions separate two GL_m-varieties—specifically, the Chow variety and the higher secant variety of the Veronese variety—while occurrence obstructions fail.
• The authors prove that the coordinate ring of the Chow variety contains certain irreducible representations with non-zero multiplicity, while the coordinate ring of the Waring rank variety does not, establishing a strict separation via multiplicities.
• A generalized Hermite reciprocity theorem is proven, which resolves Foulkes' conjecture for a new infinite family of cases, extending known results in symmetric function theory.
• The paper shows that the multiplicity obstruction criterion is strictly stronger than the occurrence obstruction criterion in this setting, answering Scott Aaronson's question affirmatively.
• The separation is achieved in a surprisingly simple and natural geometric setting: the variety of products of linear forms versus the variety of polynomials of bounded border Waring rank.
• The results demonstrate that representation-theoretic multiplicities in coordinate rings can provide stronger separation tools than mere occurrence conditions in geometric complexity theory.