Reduced Kronecker Coefficients and Counter–Examples to Mulmuley’s Strong Saturation Conjecture SH

Briand, Emmanuel; Orellana, Rosa; Rosas, Mercedes

doi:10.1007/s00037-009-0279-z

Cited by 37 publications

(46 citation statements)

References 23 publications

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“…This new description allowed us to test several conjectures of Mulmuley. As a result, we found a counterexample [4] for the strong version of his SH conjecture [16] on the behavior of the Kronecker coefficients under stretching of its indices.…”

Section: Introductionmentioning

confidence: 98%

The stability of the Kronecker product of Schur functions

2011

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In the late 1930s Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.

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Section: Introductionmentioning

confidence: 98%

The stability of the Kronecker product of Schur functions

2011

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“…In particular, these coefficients have been linked to Geometric Complexity Theory (see [2,12]), stable representation theory and FI-modules (see [5,15]) and to the study of the Fourier-Deligne transform (see [9]). …”

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confidence: 99%

Deligne categories and reduced Kronecker coefficients

Entova-Aizenbud

2016

J Algebr Comb

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The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions λ, μ, τ of n, the multiplicity of λ in μ ⊗ τ is called the Kronecker coefficient g λ μ,τ . When the first part of each of the partitions is taken to be very large (the remaining parts being fixed), the values of the appropriate Kronecker coefficients stabilize; the stable value is called the reduced (or stable) Kronecker coefficient. These coefficients also generalize the Littlewood-Richardson coefficients and have been studied quite extensively. In this paper, we show that reduced Kronecker coefficients appear naturally as structure constants of Deligne categories Rep(S t ). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories Rep(S t ) and derive new combinatorial identities.

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“…With Valiant's theory of determinant computations as its starting point, their approach relies, among other things, on the (conjectural) ability to decide in polynomial time the positivity of Kronecker coefficients and their plethystic generalizations. Envisioned as a far reaching mathematical program requiring over 100 years to complete [F2], this approach led to a flurry of activity in an attempt to understand and establish some critical combinatorial and computational properties of Kronecker coefficients (see [BOR1,BI1,CDW,Ike,M1]). This paper is a new advance in this direction.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Mulmuley's original approach to KP was via Conjecture 3.4 and a weak version of the Saturation Conjecture. The original version of the latter was disproved in [BOR1], and modified by Mulmuley in the appendix to [BOR1] (see also [M1]). …”

Section: ])mentioning

confidence: 99%

On the complexity of computing Kronecker coefficients

Pak¹,

Panova

2015

comput. complex.

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Abstract. We study the complexity of computing Kronecker coefficients g(λ, µ, ν). We give explicit bounds in terms of the number of parts ℓ in the partitions, their largest part size N and the smallest second part M of the three partitions. When M = O(1), i.e. one of the partitions is hook-like, the bounds are linear in log N , but depend exponentially on ℓ. Moreover, similar bounds hold even when M = e O(ℓ) . By a separate argument, we show that the positivity of Kronecker coefficients can be decided in O(log N ) time for a bounded number ℓ of parts and without restriction on M . Related problems of computing Kronecker coefficients when one partition is a hook, and computing characters of Sn are also considered.

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Reduced Kronecker Coefficients and Counter–Examples to Mulmuley’s Strong Saturation Conjecture SH

Cited by 37 publications

References 23 publications

The stability of the Kronecker product of Schur functions

The stability of the Kronecker product of Schur functions

Deligne categories and reduced Kronecker coefficients

On the complexity of computing Kronecker coefficients

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