Unimodality via Kronecker products

Pak, Igor; Panova, Greta

doi:10.1007/s10801-014-0520-y

“…Here we give the proof of Theorem 6.1. In order to prove this theorem, we will derive a simple formula for these Kronecker coefficients, following the approaches set in [Bla12,Liu14,PP14b]. For brevity we set g k (d, n) = g((nd − k, 1 k ), d × n, d × n).…”

Section: Exact Results For Kronecker Coefficientsmentioning

confidence: 99%

Rectangular Kronecker coefficients and plethysms in geometric complexity theory

Ikenmeyer

¹

,

Panova

²

2017

Advances in Mathematics

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We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial.Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition. We also compare rectangular Kronecker coefficients with their corresponding plethysm coefficients, which leads to a new lower bound for rectangular Kronecker coefficients. Moreover, we prove that the saturation of the rectangular Kronecker semigroup is trivial, we show that the rectangular Kronecker positivity stretching factor is 2 for a long first row, and we completely classify the positivity of rectangular limit Kronecker coefficients that were introduced by Manivel in 2011.

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“…In this case the generating function G(q) can be computed explicitly summarized in the following While the rectangles so far have been the same, we observe that in most cases when λ = (a b ) and µ = (c d ) are two different rectangles and ν = (n − k, k) is a two-row, then the Kronecker coefficients is almost always 0. As in [PP14b], the Jacobi-Trudi identity for a two row gives…”

Section: Here Strict Unimodality Meansmentioning

confidence: 90%

“…In order to prove this theorem, we will derive a simple formula for these Kronecker coefficients, following the approaches set in [Bla12,Liu14,PP14b]. For brevity we set…”

mentioning

confidence: 99%

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Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

Ikenmeyer

¹

,

Panova

²

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial.Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition. We also compare rectangular Kronecker coefficients with their corresponding plethysm coefficients, which leads to a new lower bound for rectangular Kronecker coefficients. Moreover, we prove that the saturation of the rectangular Kronecker semigroup is trivial, we show that the rectangular Kronecker positivity stretching factor is 2 for a long first row, and we completely classify the positivity of rectangular limit Kronecker coefficients that were introduced by Manivel in 2011.

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“…8.5. A positive combinatorial interpretation for the Kronecker coefficients, analogous to the LR-rule, would likely show that the decision problem is in NP and the counting problem in #P. Such interpretation would also imply a combinatorial interpretation for the difference between the number of partitions of k and the number of partitions of k − 1, which fit into an ℓ × m rectangle (see [PP1]). Formally, this difference is equal to g m ℓ , m ℓ , (n − k, k) ; in full generality its combinatorial interpretation is already highly nontrivial and will appear in [PP3] (see also [BO]).…”

Section: ])mentioning

confidence: 99%

On the complexity of computing Kronecker coefficients

Pak¹,

Panova

²

2015

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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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Unimodality via Kronecker products

Cited by 24 publications

References 31 publications

Rectangular Kronecker coefficients and plethysms in geometric complexity theory

Rectangular Kronecker coefficients and plethysms in geometric complexity theory

Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

On the complexity of computing Kronecker coefficients

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