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Using the framework of the quantum separation of variables (SoV) for higherrank quantum integrable lattice models [1], we introduce some foundations to gobeyond the obtained complete transfer matrix spectrum description, and open theway to the computation of matrix elements of local operators. This firstamounts to obtain simple expressions for scalar products of the so-calledseparate states (transfer matrix eigenstates or some simple generalization ofthem). In the higher rank case, left and right SoV bases are expected to bepseudo-orthogonal, that is for a given SoV co-vector, there could be more thanone non-vanishing overlap with the vectors of the chosen right SoV basis. Forsimplicity, we describe our method to get these pseudo-orthogonality overlapsin the fundamental representations of the $\mathcal{Y}(gl_3)$ lattice modelwith $N$ sites, a case of rank 2. The non-zero couplings between the co-vectorand vector SoV bases are exactly characterized. While the correspondingSoV-measure stays reasonably simple and of possible practical use, we addressthe problem of constructing left and right SoV bases which do satisfy standardorthogonality. In our approach, the SoV bases are constructed by using familiesof conserved charges. This gives us a large freedom in the SoV basesconstruction, and allows us to look for the choice of a family of conservedcharges which leads to orthogonal co-vector/vector SoV bases. We first definesuch a choice in the case of twist matrices having simple spectrum and zerodeterminant. Then, we generalize the associated family of conserved charges andorthogonal SoV bases to generic simple spectrum and invertible twist matrices.Under this choice of conserved charges, and of the associated orthogonal SoVbases, the scalar products of separate states simplify considerably and take aform similar to the $\mathcal{Y}(gl_2)$ rank one case.

On Scalar Products in Higher Rank Quantum Separation of Variables