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 ${\mathrm{\Lambda }}_R$. 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 $\mathbb{Z}$.