Linearly Independent Products of Rectangularly Complementary Schur Functions

Abstract

Fix a rectangular Young diagram $R$, and consider all the products of Schur functions $s_\lambda s_\lambda^{c}$, where $\lambda$ and $\lambda^{c}$ run over all (unordered) pairs of partitions which are complementary with respect to $R$.

Theorem: The self-complementary products, $s_\lambda^2$ where $\lambda=\lambda^{c}$, are linearly independent of all other $s_\lambda s_\lambda^{c}$.

Conjecture: The products $s_\lambda s_\lambda^{c}$ are all linearly independent.