No occurrence obstructions in geometric complexity theory

Abstract

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VPws and VNP. Mulmuley and Sohoni [37] suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL n 2 (C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed in [37]. 1.3. Conjecture (Mulmuley and Sohoni 2001). For all c ∈ N ≥1 , for infinitely many m, there exists a partition λ occurring in C[Z m c ,m ] but not in C[Ω m c ].This conjecture implies Conjecture 1.2 by the above reasoning. Conjecture 1.3 on the existence of occurrence obstructions has stimulated a lot of research and has been the main focus of researchers in geometric complexity theory in the past years, see Section 1 (b). Unfortunately, this conjecture is false! This is the main result of this work. More specifically, we show the following.1.4. Theorem. Let n, d, m be positive integers with n ≥ m 25 and λ ⊢ nd. If λ occurs in C[Z n,m ], then λ also occurs in C[Ω n ]. In particular, Conjecture 1.3 is false.One can likely improve the bound on n by a more careful analysis. 1 (b). Related work: Kronecker coefficients. Kronecker coefficients are fundamental quantities that have been the object of study in algebraic combinatorics for a long time [40]. A difficulty in their study is that there is no known counting interpretation of them [42]. The Kronecker coefficient k(λ, µ, ν) of three partitions λ, µ, ν of the same size d is defined as the dimension of the, where [λ] denotes the irreducible S d -module of type λ and S d is the symmetric group on d symbols; see [32, I §7, internal product]. We write n × d for the rectangular partition (d, . . . , d) of size nd and call k n (λ) := k(λ, n × d, n × d) the rectangular Kronecker coefficient of λ ⊢ nd. In [25] it was shown that deciding positivity of Kronecker coefficients in general is NP-hard, but this proof fails for rectangular formats.Let K n (λ) denote the multiplicity by which the irreducible GL n 2 -module of type λ ⊢ nd occurs in C[Ω n ] d . We call the numbers K n (λ) the GCT-coefficients. In [37] it was realized that GCTcoefficients can be upper bounded by rectangular Kronecker coefficients: we have K n (λ) ≤ k n (λ) for λ ⊢ nd. In fact, the multiplicity of λ in the coordinate ring of the orbit GL n 2 · det n equals the so-called symmetric rectangular Kronecker coefficient [12], which is upper bounded by k n (λ).Note that an occurrence obstruction for Z n,m ⊆ Ω n is a partition λ for which K n (λ) = 0 and such that λ occurs in C[Z n,m ]. Since hardly anything was known about the...

Geometric complexity theory and matrix powering

“…Such λ are called occurrence obstructions. It was recently shown that no lower bounds better than dc(per m ) > m 25 can be proved with occurrence obstructions [BIP16]. Mulmuley and Sohoni proposed even further to use the following upper bound for a ′ λ (d[n]) coming from the coordinate ring of the determinant orbit: The algebraic group GL n 2 is an affine variety and acts on itself by left and right multiplication.…”

“…4.2.7]) tells us how its coordinate ring splits as a GL n 2 ×GL n 2 -representation: C[GL n 2 ] ≃ λ V λ ⊗V λ * , where the sum is over all isomorphism types of GL n 2 and λ * is the type dual to λ. If p ∈ A n has a reductive stabilizer S ⊆ GL n 2 , then the orbit GL n 2 p is an affine variety whose coordinate ring C[GL n 2 p] is the ring of right S-invariants: C[GL n 2 p] = C[GL n 2 ] S , see [BIP16,Sec. 4.1 & 4.2].…”

“…Its existence implies dc(per m ) > n. Mulmuley and Sohoni conjectured that orbit occurrence obstructions which prove Conj. 2.1 exist, recently disproved in [BIP16]. A natural upper bound for sk(λ, n × d) is the Kronecker coefficient g(λ, n × d, n × d).…”

“…The results in [IP16] (g(λ, n × d, n × d) > 0) and [BIP16] (a ′ λ (d[n]) > 0) are proved using the semigroup property in the following way: One decomposes λ into a sum of smaller partitions, then shows positivity for the smaller partitions, and then uses the semigroup property. In both papers Prop.…”

Waring Rank, Parameterized and Exact Algorithms

“…The study of Waring rank is a classical problem in algebraic geometry and invariant theory, with pioneering work done in the second half of the 19th century by A. Clebsch, J.J. Sylvester, and T. Reye, among others [IK99,Introduction]. It has enjoyed a recent resurgence of popularity within algebraic geometry [IK99,Lan12] and has connections in computer science to the limiting exponent of matrix multiplication ω [CHI + 18], the Mulmuley-Sohoni Geometric Complexity Theory program [BIP19], and several other areas in algebraic complexity [Lan17,EGOW18]. This paper adds parameterized algorithms to this list, showing that several methods in this area (color-coding methods [AYZ95, AG07, HWZ08], the group-algebra/determinant sum approach [Kou08, Wil09,Bjö10], and inclusion-exclusion methods) fundamentally result from rank upper bounds for a specific family of polynomials.…”