An explicit characterization of socular simple modules of $\mathfrak{sl}(n,\mathbb{C})$

TL;DR

This paper provides an explicit characterization of socular simple modules in the parabolic category O for rak{sl}(n,\u00C4) using highest weights and Young tableaux, clarifying their structure and properties.

Contribution

It offers a new explicit description of socular simple modules in rak{sl}(n,\u00C4) based on highest weights and Young tableaux, advancing understanding of their structure.

Findings

Characterization given in terms of highest weight and Young tableau.

Clarifies the structure of socular simple modules in rak{sl}(n,\u00C4).

Links module properties to combinatorial data.

Abstract

Let $\mathfrak{g}$ be a simple complex Lie algebra with a Cartan subalgebra $\mathfrak{h}$. We fix a standard parabolic subalgebra $\mathfrak{p}\supset \mathfrak{h}$. The socular simple modules play an important role in the parabolic versions of category $\mathcal{O}^{\mathfrak{p}}$. From Irving's work, we know that these modules are just those modules with largest possible Gelfand-Kirillov dimension in $\mathcal{O}^{\mathfrak{p}}$. In this article, we will give an explicit characterization for these modules of $\mathfrak{sl}(n,\mathbb{C})$. Our characterization is given in the information of the corresponding highest weight and Young tableau.