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

Critical classes, Kronecker products of spin characters, and the Saxl conjecture

Highlighting the use of critical classes, we consider constituents in Kronecker products, in particular of spin characters of the double covers of the symmetric and alternating groups. We apply results from the spin case to find constituents in Kronecker products of characters of the symmetric groups. Via this tool, we make progress on the Saxl conjecture; this claims that for a triangular number $n$, the square of the irreducible character of the symmetric group $S_n$ labelled by the staircase contains all irreducible characters of $S_n$ as constituents. We find a large number of constituents in this square which were not detected by other methods. Moreover, the investigation of Kronecker products of spin characters inspires a spin variant of Saxl's conjecture.Comment: 17 page

Saxl conjecture and the tensor square of unipotent characters of GL(n,q)

We know from Letellier that if for some triple of partitions the corresponding Kronecker coefficient is non-zero then the corresponding multiplicities for unipotent characters of GL(n,q) is also non-zero. A conjecture of Saxl says that the tensor square of an irreducible character of the symmetric group corresponding to a staircase partition contains all the irreducible characters. Therefore Saxl conjecture implies its analogue for unipotent characters. In this paper we prove the analogue of Saxl conjecture for unipotent characters and we describe conjecturally the set of all partitions for which the tensor square of the associated unipotent character contains all the unipotent characters

Covering Irrep(Sn)Irrep(S_n) With Tensor Products and Powers

We study when a tensor product of irreducible representations of the symmetric group $S_n$ contains all irreducibles as subrepresentations - we say such a tensor product covers $Irrep(S_n)$. Our results show that this behavior is typical. We first give a general criterion for such a tensor product to have this property. Using this criterion we show that the tensor product of a constant number of random irreducibles covers $Irrep(S_n)$ asymptotically almost surely. We also consider, for a fixed irreducible representation, the degree of tensor power needed to cover $Irrep(S_n)$. We show that the simple lower bound based on dimension is tight up to a universal constant factor for every irreducible representation, as was recently conjectured by Liebeck, Shalev, and Tiep

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

On the Kronecker product of Schur functions of square shapes

Motivated by the Saxl conjecture and the tensor square conjecture, which states that the tensor squares of certain irreducible representations of the symmetric group contain all irreducible representations, we study the tensor squares of irreducible representations associated with square Young diagrams. We give a formula for computing Kronecker coefficients, which are indexed by two square partitions and a three-row partition, specifically one with a short second row and the smallest part equal to 1. We also prove the positivity of square Kronecker coefficients for particular families of partitions, including three-row partitions and near-hooks.Comment: 23 page