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We study the $\mathfrak{gl}_{1|1}$ supersymmetric XXX spin chains. We give anexplicit description of the algebra of Hamiltonians acting on any cyclic tensorproducts of polynomial evaluation $\mathfrak{gl}_{1|1}$ Yangian modules. Itfollows that there exists a bijection between common eigenvectors (up toproportionality) of the algebra of Hamiltonians and monic divisors of anexplicit polynomial written in terms of the Drinfeld polynomials. In particularour result implies that each common eigenspace of the algebra of Hamiltonianshas dimension one. We also give dimensions of the generalized eigenspaces. Weshow that when the tensor product is irreducible, then all eigenvectors can beconstructed using Bethe ansatz. We express the transfer matrices associated tosymmetrizers and anti-symmetrizers of vector representations in terms of thefirst transfer matrix and the center of the Yangian.

On the supersymmetric XXX spin chains associated to $\mathfrak{gl}_{1|1}$