The immersion poset on partitions

Abstract

We introduce the immersion poset $(\mathcal{P}(n),{⩽}_I)$ on partitions, defined by $\hat{I}»{⩽}_I\hat{I}¼$ if and only if $s_{\hat{I}}¼(x_1,\dots ,x_N)-s_{\hat{I}}»(x_1,\dots ,x_N)$ is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of $G{L}_N(\mathbb{C})$ form an immersion pair, as defined by Prasad and Raghunathan (2022). We develop injections $\mathsf{SSYT}(\hat{I}»,\hat{I}½)\hookrightarrow \mathsf{SSYT}(\hat{I}¼,\hat{I}½)$ on semistandard Young tableaux given constraints on the shape of $\hat{I}»$, and present results on immersion relations among hook and two column partitions. The standard immersion poset $(\mathcal{P}(n),{⩽}_{std})$ is a refinement of the immersion poset, defined by $\hat{I}»{⩽}_{std}\hat{I}¼$ if and only if $\hat{I}»{⩽}_D\hat{I}¼$ in dominance order and $f^{\hat{I}}»⩽f^{\hat{I}}¼$, where $f^{\hat{I}}½$ is the number of standard Young tableaux of shape $\hat{I}½$. 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 $p_{A_{\hat{I}}¼}$ on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram (2018).

34 pages