66 research outputs found

    Rectangular Kronecker coefficients and plethysms in geometric complexity theory

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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.Comment: 20 page

    Even Partitions in Plethysms

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    We prove that for all natural numbers k,n,d with k <= d and every partition lambda of size kn with at most k parts there exists an irreducible GL(d, C)-representation of highest weight 2*lambda in the plethysm Sym^k(Sym^(2n) (C^d)). This gives an affirmative answer to a conjecture by Weintraub (J. Algebra, 129 (1):103-114, 1990). Our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory.Comment: 9 page

    No occurrence obstructions in geometric complexity theory

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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 VP_{ws} and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) 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 by Mulmuley and Sohoni.Comment: Substantial revision. This version contains an overview of the proof of the main result. Added material on the model of power sums. Theorem 4.14 in the old version, which had a complicated proof, became the easy Theorem 5.4. To appear in the Journal of the AM

    Obstructions to combinatorial formulas for plethysm

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    Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of S3(Sk)S^3(S^k) and Sk(S3)S^k(S^3), that these need not be counting functions of inhomogeneous polytopes of dimension equal to the degree of the quasi-polynomial. It follows that these functions are not, in general, counting functions of lattice points in any scaled convex bodies, even when restricted to single rays. Our results also apply to special rectangular Kronecker coefficients.Comment: 7 pages; v2: Improved version with further reaching counterexamples; v3: final version as in Electronic Journal of Combinatoric

    The stability of the Kronecker products of Schur functions

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    In the late 1930's 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.Comment: 16 page

    Rectangular symmetries for coefficients of symmetric functions

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    We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.Ministerio de Ciencia e InnovaciónJunta de AndalucíaNational Science Foundation Gran

    Breaking down the reduced Kronecker coefficients

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    We resolve three interrelated problems on \emph{reduced Kronecker coefficients} g(α,β,γ)\overline{g}(\alpha,\beta,\gamma). First, we disprove the \emph{saturation property} which states that g(Nα,Nβ,Nγ)>0\overline{g}(N\alpha,N\beta,N\gamma)>0 implies g(α,β,γ)>0\overline{g}(\alpha,\beta,\gamma)>0 for all N>1N>1. Second, we esimate the maximal g(α,β,γ)\overline{g}(\alpha,\beta,\gamma), over all α+β+γ=n|\alpha|+|\beta|+|\gamma| = n. Finally, we show that computing g(λ,μ,ν)\overline{g}(\lambda,\mu,\nu) is strongly #P\# P-hard, i.e. #P\#P-hard when the input (λ,μ,ν)(\lambda,\mu,\nu) is in unary.Comment: 5 page
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