The \mathfrak{gl}_{1|1} divisor parametrization conjecture for arbitrary modules

Let $\boldsymbol\lambda$ be a sequence of arbitrary $\mathfrak{gl}_{1|1}$ weights, and let $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$ be its singular subspace. Let $e_{21}^{\boldsymbol s}$ denote the relevant lowering operator, and let $\mathcal N(T)$ be the polynomial associated with the module data. Form the subspace of $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$ spanned by the joint eigenvectors of the Gaudin Hamiltonians $\mathcal H_k$, and quotient it by its intersection with the image of $e_{21}^{\boldsymbol s}$. Arbitrary-module divisor conjecture. On this subquotient, the Gaudin Hamiltonians $\mathcal H_k$, $k=1,\dots,n$, have a simple joint spectrum, and their joint eigenvectors of weight $(p-l,q+l)$, up to multiplication by a nonzero constant, are in one-to-one correspondence with monic divisors $y$ of $\mathcal N(T)$ of degree $l$. Under this correspondence, $\mathcal H_kv=E_kv$ for $k=1,\dots,n$, where $E_k$ are given by the stated formula. This modifies the polynomial-module conjecture to account for non-complete reducibility and identifies eigenvectors differing through the image of $e_{21}^{\boldsymbol s}$; it is not proved in the paper.