Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

Ikenmeyer, Christian; Panova, Greta

doi:10.1109/focs.2016.50

Cited by 17 publications

(28 citation statements)

References 37 publications

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“…For this reason, K. Mulmuley suggested that we investigate the barriers of the method of shifted partials in the unpadded setting (although Mulmuley himself anticipated our answer). Similar concerns were shown after the results of [IP17] and [BIP19] on the Geometric Complexity Theory program, and were partially addressed in [GIP17], exploiting the same homogenization result of [Nis91] that we use in this work.…”

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confidence: 61%

Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley

Gesmundo,

Landsberg

2017

Preprint

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We answer a question of K. Mulmuley: In [ELSW18] it was shown that the method of shifted partial derivatives cannot be used to separate the padded permanent from the determinant. Mulmuley asked if this "no-go" result could be extended to a model without padding. We prove this is indeed the case using the iterated matrix multiplication polynomial. We also provide several examples of polynomials with maximal space of partial derivatives, including the complete symmetric polynomials. We apply Koszul flattenings to these polynomials to have the first explicit sequence of polynomials with symmetric border rank lower bounds higher than the bounds attainable via partial derivatives.

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confidence: 61%

Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley

Gesmundo,

Landsberg

2017

Preprint

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“…Via the standard projection C n×n → C m×m , we can view X n−m 11 per m as an element of the bigger space ∈ Sym n (C n×n ) * . (Sometimes the padding is achieved by using a linear form different from X 11 , e.g., but this is irrelevant, see [26,Appendix]. )…”

Section: Introductionmentioning

confidence: 99%

No occurrence obstructions in geometric complexity theory

Bürgisser

Ikenmeyer

Panova

2018

J. Amer. Math. Soc.

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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...

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“…Proof. This is proved by a finite calculation for all cases but (3, 3) as Thm 1.10(a) in [IP17]. Exactly the same calculation can be used to also prove the result for the additional partition (3, 3).…”

Section: No Occurrence Obstructionsmentioning

confidence: 63%

“…The papers [IP17, BIP19] rule out occurrence obstructions for families, but only in ranges where they would give very strong new algebraic circuit lower bounds, so that we expect it to be difficult to find multiplicity obstructions in those cases. Note also that [IP17,BIP19] are only dealing with padded polynomials, for which [KL14] guarantees λ to have a very restricted shape.…”

Section: Representation Theoretic Obstructionsmentioning

confidence: 99%

“…At the heart of the approach was the hope that so-called occurrence obstructions (see Section 2) would be sufficient to separate VP and VNP. In [IP17,BIP19] it was shown that occurrence obstructions are too weak to provide the necessary separation, at least for the group varieties that were originally proposed by Mulmuley and Sohoni. But representation theoretic multiplicities might still be able to separate VP and VNP when we look at the finer separation criterion via multiplicity obstructions (see also Section 2).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions

Dörfler¹,

Ikenmeyer²,

Panova³

2019

Preprint

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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 Bürgisser-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.

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Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

Cited by 17 publications

References 37 publications

Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley

Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley

No occurrence obstructions in geometric complexity theory

On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions

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