On exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chains

We study the integrable two-site states of the quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(N)$-invariant R-matrix. We investigate the overlaps between the integrable two-site states and the wave-functions. To find exact derivations for the factorized overlap formulas for the nested integrable systems is a longstanding unsolved problem. In this paper we give a derivation for a large class of the integrable states of the $\mathfrak{gl}(N)$ symmetric spin chain. The first part of the derivation is to calculate recurrence relations for the off-shell overlap that uniquely fix it. Using these recursions we prove that the normalized overlaps of the multi-particle states have factorized forms which contain the products of the one-particle overlaps and the ratio of the Gaudin-like determinants. We also show that the previously proposed overlap formulas agree with our general formula.