Twists and Singular Vectors in $\widehat{s\ell}(2|1)$ Representations

We propose new formulas for singular vectors in Verma modules over the affine Lie superalgebra $\widehat{s\ell}(2|1)$. We analyze the coexistence of singular vectors of different types and identify the twisted modules arising as submodules and quotient modules of $\widehat{s\ell}(2|1)$ Verma modules. We show that with the twists (spectral flow transformations) properly taken into account, a resolution of irreducible representations can be constructed consisting of only the $\mathscr N_{h,k;\theta}^\pm$ modules.