Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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.
['Michael Kleber']
I am writing a research paper on a idea Linearly Independent Products of Rectangularly Complementary Schur Functions
[{"from": "human", "value": "I am writing a research paper on a idea Linearly Independent Products of Rectangularly Complementary Schur Functions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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."}]