Odd singular vector formula for general linear Lie superalgebras

TL;DR

This paper derives a closed-form formula for a specific singular vector in Verma modules of the general linear Lie superalgebra, especially for atypical weights, and proves its uniqueness and descent to Kac modules.

Contribution

It provides the first explicit formula for odd singular vectors in Verma modules of (m|n) for atypical weights, clarifying their structure and uniqueness.

Findings

Explicit formula for odd singular vectors in (m|n) Verma modules.

Proof of uniqueness of these singular vectors up to scalar multiples.

Demonstration that vectors descend to Kac modules under certain conditions.

Abstract

We establish a closed formula for a singular vector of weight $\lambda-\beta$ in the Verma module of highest weight $\lambda$ for Lie superalgebra $\mathfrak{gl}(m|n)$ when $\lambda$ is atypical with respect to an odd positive root $\beta$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $\lambda$ and $\lambda-\beta$ are dominant integral.