Staircase Minimality and a Proof of Saxl's Conjecture
Soong Kyum Lee
TL;DR
The paper settles Saxl's conjecture unconditionally by proving a Staircase Minimality Theorem: among all 2-regular partitions of the triangular size $T_k$, the staircase $\rho_k$ is the unique dominance-minimal element. This enables a dominance-based positivity via Ikenmeyer's result, and, crucially, modular saturation is achieved using only diagonal decomposition data, after which the Bessenrodt–Bowman–Sutton lifting completes the positivity. Consequently, every $\lambda \vdash T_k$ appears in the tensor square $S^{\rho_k} \otimes S^{\rho_k}$, establishing Kronecker universality for staircases at triangular numbers. The work also characterizes Kronecker-universal self-conjugate partitions, showing staircases are the sole examples at triangular $n$, and provides illustrative examples and open questions on extending these ideas to broader regularity settings.
Abstract
Saxl's conjecture (2012) asserts that for the staircase partition $ρ_k = (k, k-1, \ldots, 1)$, 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 $ρ_k$ 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 $d_{μμ} = 1$ 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.