What is in #P and what is not?

Ikenmeyer, Christian; Pak, Igor

doi:10.1109/focs54457.2022.00087

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Final Remarks and Open Problems1

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2023

2024

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Cited by 6 publications

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“…This was a major motivation for our investigation in [10]. We show in [26,Section 7.4] that the AD inequality (5-3) does not have a combinatorial interpretation in full generality, in a sense of being in #P. Of course, the Lam-Postnikov-Pylyavskyy deep algebraic approach in [30] (see Remark 4.2) is even less likely to give a combinatorial interpretation. We refer to [33,Section 6] for an extensive survey.…”

Section: Final Remarks and Open Problemsmentioning

confidence: 98%

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2023

Pacific J. Math.

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Section: Final Remarks and Open Problemsmentioning

confidence: 98%

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2023

Pacific J. Math.

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Equality cases of the Alexandrov–Fenchel inequality are not in the polynomial hierarchy

Chan,

Pak

2024

Forum of Mathematics, Pi

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Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

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All Kronecker coefficients are reduced Kronecker coefficients

Ikenmeyer,

Panova

2024

Forum of Mathematics, Pi

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We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of an open problem by Stanley from 2000 and an open problem by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is ${\textsf {NP}}$ -hard, and computing them is ${{{\textsf {#P}}}}$ -hard under parsimonious many-one reductions. Our proof also provides an explicit isomorphism of the corresponding highest weight vector spaces.

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What is in #P and what is not?

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Cited by 6 publications

References 126 publications

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Equality cases of the Alexandrov–Fenchel inequality are not in the polynomial hierarchy

All Kronecker coefficients are reduced Kronecker coefficients

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