The immersion poset on partitions

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).