Saxl's conjecture: the tensor square of the staircase representation contains every irreducible representation of the symmetric group

Theorem 1.1: for every $m\ge1$ and every $\lambda\vdash N_m=m(m+1)/2$, $g(\rho_m,\rho_m,\lambda)>0$, so the tensor square of the staircase Specht module contains every irreducible representation of $S_{N_m}$. The proof gives the stronger statement that every constituent occurs inside one explicitly generated cyclic submodule of the tensor square, extending Ikenmeyer's triangular arrangement of alternating tensors. Not shown: values or bounds for these Kronecker coefficients, or a combinatorial interpretation of them.

Theorem 1.1: for every $m\ge1$ and every $\lambda\vdash N_m=m(m+1)/2$, $g(\rho_m,\rho_m,\lambda)>0$, so the tensor square of the staircase Specht module contains every irreducible representation of $S_{N_m}$. The proof gives the stronger statement that every constituent occurs inside one explicitly generated cyclic submodule of the tensor square, extending Ikenmeyer's triangular arrangement of alternating tensors. Not shown: values or bounds for these Kronecker coefficients, or a combinatorial interpretation of them.