The immersion poset on partitions

TL;DR

This work defines and analyzes the immersion poset on partitions, linking monomial-positivity of Schur polynomial differences to immersion relations among polynomial GL$_N$-representations. It introduces a refined standard immersion poset based on dominance and standard tableau counts, and establishes explicit maximality criteria across shapes, including hooks and two-column partitions. The authors develop concrete injections between sets of semistandard Young tableaux to study immersion covers, and they characterize immersion relations in hooks and two-column cases, providing a detailed picture of the poset's structure. They also investigate lower intervals, proving Schur-positivity for several conjectured interval power sums (notably $p_{[(1^n),(n-2,2)]}$ and $p_{[(1^n),(n-2,1,1)]}$ for large $n$), thereby connecting representation-theoretic questions to symmetric-function positivity and ribbon-tableaux combinatorics, with open questions on maximal elements, interval classifications, and asymptotic behavior.

Abstract

We introduce the immersion poset $(\mathcal{P}(n), \leqslant_I)$ on partitions, defined by $λ\leqslant_I μ$ if and only if $s_μ(x_1, \ldots, x_N) - s_λ(x_1, \ldots, x_N)$ 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 $\mathsf{SSYT}(λ, ν) \hookrightarrow \mathsf{SSYT}(μ, ν)$ on semistandard Young tableaux given constraints on the shape of $λ$, 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 $λ\leqslant_{std} μ$ if and only if $λ\leqslant_D μ$ in dominance order and $f^λ\leqslant f^μ$, where $f^ν$ is the number of standard Young tableaux of shape $ν$. 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 $p_{A_μ}$ on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram (2018).

Figures (3)

• Figure 1: The (standard) immersion poset for $n=8$.
• Figure 2: Immersion poset restricted to hook partitions for $n=2k+1$ (left) and $n=2k$ (right).
• Figure 3: Subposet of the immersion poset only containing partitions in $A_{(n-2,2)}$ for $n=8$ (left), $n=9$ (middle), and $n=15$ (right).

Theorems & Definitions (87)

• Definition 1.2
• Lemma 2.1
• proof
• Definition 2.2
• Example 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Remark 2.6