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 [7]. We develop injections \(\textsf{SSYT}(\lambda , \nu ) \hookrightarrow \textsf{SSYT}(\mu , \nu )\) 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 on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram [12].
