The Saxl Conjecture and the Dominance Order

In 2012 Jan Saxl conjectured that all irreducible representations of the symmetric group occur in the decomposition of the tensor square of the irreducible representation corresponding to the staircase partition. We make progress on this conjecture by proving the occurrence of all those irreducibles which correspond to partitions that are comparable to the staircase partition in the dominance order. Moreover, we use our result to show the occurrence of all irreducibles corresponding to hook partitions. This generalizes results by Pak, Panova, and Vallejo from 2013.Comment: 11 page

On McKay's propagation theorem for the Foulkes conjecture

We translate the main theorem in Tom McKay's paper "On plethysm conjectures of Stanley and Foulkes" (J. Alg. 319, 2008, pp. 2050-2071) to the language of weight spaces and projections onto invariant spaces of tensors, which makes its proof short and elegant.Comment: 9 page

Binary Determinantal Complexity

We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite graphs. Furthermore, we analyze sequences of polynomials that are determinants of polynomially sized matrices consisting only of zeros, ones, and variables. We show that these are exactly the sequences in the complexity class of constant free polynomially sized (weakly) skew circuits.Comment: 10 pages, C source code for the computation available as ancillary file

Rectangular Kronecker coefficients and plethysms in geometric complexity theory

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

No occurrence obstructions in geometric complexity theory

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