Staircase Minimality and a Proof of Saxl's Conjecture

Saxl's conjecture (2012) asserts that for the staircase partition $\rho_k = (k, k-1, \ldots, 1)$, the tensor square of the corresponding irreducible representation of the symmetric group $S_{T_k}$ contains every irreducible representation as a constituent, where $T_k = k(k+1)/2$ is the $k$th triangular number. We prove this conjecture unconditionally. Our proof introduces the Staircase Minimality Theorem: among all 2-regular partitions of $T_k$, the staircase $\rho_k$ is the unique dominance-minimal element. Combined with Ikenmeyer's theorem on dominance and Kronecker positivity for staircases, this establishes that every 2-regular partition appears in the tensor square. Modular saturation then follows using only the diagonal entries $d_{\mu\mu} = 1$ of the decomposition matrix, and the Bessenrodt--Bowman--Sutton lifting theorem completes the proof. We further prove that at triangular numbers, staircases are the only Kronecker-universal self-conjugate partitions, providing a complete characterization.