Kleber's conjecture and complementary products of symmetric functions

We prove Kleber's rectangular-complement conjecture for Schur functions over an arbitrary commutative ring $R$, showing that, for a fixed rectangle, the products $s_\lambda s_{\lambda^\vee}$, indexed by unordered complementary pairs, are linearly independent in $\Lambda_R$. The proof rests on a general independence theorem for componentwise splittings, which asserts that for every partition $\theta$, the products $s_\alpha s_\beta$ are linearly independent as $\{\alpha,\beta\}$ ranges over unordered pairs of partitions satisfying $\alpha+\beta=\theta$. The independence of the products $s_\lambda s_{\lambda^\vee}$ also yields linear independence of the Koike--Terada universal-character products over any field, answering a question of Gao--Orelowitz--Yong. We also prove the analogous result for monomial symmetric functions over fields of characteristic zero, as well as integral linear independence over $\mathbb{Z}$.