An explicit characterization of socular simple modules of $\mathfrak{sl}(n,\mathbb{C})$
- Resource Type
- Working Paper
- Authors
- Bai, Zhanqiang; Xiao, Wei; Xie, Xun
- Source
- Subject
- Mathematics - Representation Theory
- Language
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.
Comment: Similar result appeared in some other papers