OAI.Saxl.saxl_conjecture

The theorem states, as an admitted result, the Saxl conjecture for staircase partitions: for every natural number m ≥ 1 and every Young diagram μ with the same number of cells as the staircase diagram of size m (the diagram of cells (i,j) with i,j < m and i+j < m, so m(m+1)/2 cells), the Kronecker multiplicity of μ in the square of the staircase is positive. Here the staircase Specht module is the complex representation of the symmetric group on n = m(m+1)/2 positions spanned by the orbit of a polytabloid, namely the signed sum over the column-preserving permutations of a tableau acting on the row tabloid, realized inside functions on words. Its Kronecker multiplicity with μ is the complex dimension of the space of symmetric-group-equivariant maps from the Specht module of μ into the tensor product of the staircase Specht module with itself, with the diagonal action. So every irreducible Specht module of that size occurs at least once in the tensor square of the staircase representation, using fixed canonical tableaux for each diagram.