Exact overlaps for all integrable two-site boundary states of $\mathfrak{gl}(N)$ symmetric spin chains

We find closed formulas for the overlaps of Bethe eigenstates of $\mathfrak{gl}(N)$ symmetric spin chains and integrable boundary states. We derive the general overlap formulas for $\mathfrak{gl}(M)\oplus\mathfrak{gl}(N-M)$ symmetric boundary states and give a well-established conjecture for the $\mathfrak{sp}(N)$ symmetric case. Combining these results with the previously derived $\mathfrak{so}(N)$ symmetric formula, now we have the overlap functions for all integrable boundary states of the $\mathfrak{gl}(N)$ spin chains which are built from two-site states. The calculations are independent from the representations of the quantum space therefore our formulas can be applied for the $SO(6)$ and the alternating $SU(4)$ spin chains which describe the scalar sectors of $\mathcal{N}=4$ super Yang-Mills and ABJM theories which are important application areas of our results.