On Bethe eigenvectors and higher transfer matrices for supersymmetric spin chains

We study the ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$ XXX spin chains defined on tensor products of highest ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$-modules. We show that the on-shell Bethe vectors are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues, confirming Conjecture 5.15 of arXiv:2007.15573 and extending the main results of arXiv:0605015 to supersymmetric case. We then take the classical limits and obtain the corresponding results for the ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$ Gaudin models.