The problem of deciding the positivity of Kronecker coefficients and Saxl conjecture

Given an positive integer $k$, let $n:=\left(\genfrac{}{}{0px}{}{k+1}{2}\right)$. In 2012, during a talk at UCLA, Jan Saxl conjectured that all irreducible representations of the symmetric group $S_n$ occur in the decomposition of the tensor square of the irreducible representation corresponding to the staircase partition. In this paper, we investigate two useful methods to obtain some irreducible representations that occur in this decomposition. Our main tolls are the semi-group property for Kronecker coefficients and generalized blocks of symmetric groups.