= Hall inner product of symmetric functions
{c}
{title2=$\langle p_\alpha,p_\beta\rangle=\delta_{\alpha\beta}z_\alpha$}
Here $p_\alpha$ is a product of power-sum <symmetric functions> and $z_\alpha=\prod_r r^{m_r}m_r!$ for part multiplicities $m_r$. The <Frobenius characteristic map> identifies this product with the <ordinary character> <inner product>. <Schur functions> are orthonormal, and the adjoint of multiplication by $p_k$ is $k\partial/\partial p_k$. These properties turn a power-sum multiplication formula into the <Murnaghan–Nakayama rule>.
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