  We introduce the immersion poset $(\mathcal{P}(n), \leqslant_I)$ on
partitions, defined by $\lambda \leqslant_I \mu$ if and only if $s_\mu(x_1,
\ldots, x_N) - s_\lambda(x_1, \ldots, x_N)$ is monomial-positive. Relations in
the immersion poset determine when irreducible polynomial representations of
$GL_N(\mathbb{C})$ form an immersion pair, as defined by Prasad and Raghunathan
(2022). We develop injections $\mathsf{SSYT}(\lambda, \nu) \hookrightarrow
\mathsf{SSYT}(\mu, \nu)$ on semistandard Young tableaux given constraints on
the shape of $\lambda$, and present results on immersion relations among hook
and two column partitions. The standard immersion poset $(\mathcal{P}(n),
\leqslant_{std})$ is a refinement of the immersion poset, defined by $\lambda
\leqslant_{std} \mu$ if and only if $\lambda \leqslant_D \mu$ in dominance
order and $f^\lambda \leqslant f^\mu$, where $f^\nu$ is the number of standard
Young tableaux of shape $\nu$. We classify maximal elements of certain shapes
in the standard immersion poset using the hook length formula. Finally, we
prove Schur-positivity of power sum symmetric functions $p_{A_\mu}$ on
conjectured lower intervals in the immersion poset, addressing questions posed
by Sundaram (2018).
