Separation of Variables for Rational $$\mathfrak {gl}(\mathsf {n})$$ Spin Chains in Any Compact Representation, via Fusion, Embedding Morphism and Bäcklund Flow

We propose a way to separate variables in rational integrable $\mathfrak{gl}(n)$ spin chain with an arbitrary finite-dimensional irreducible representation at each site and generic twisted periodic boundary conditions. Firstly, we construct basis that diagonalises higher-rank version of the Sklyanin B-operator; construction is based on recursive usage embedding $\mathfrak{gl}(k)$ into $\mathfrak{gl}(k+1)$ which induced from Yangian homomorphism controlled by dual diagonals Gelfand-Tsetlin patterns. Then, show same can be equivalently constructed action Backlund-transformed fused transfer matricies, whence Bethe wave functions factorise product ascending Slater determinants Baxter Q-functions. Finally, raising lowering operators -- conjugate momenta as normal-ordered Wronskian expressions Q-operators evaluated zeros B separated variables. It immediate consequence proposed algebra comprises maximal possible number mutually commuting charges necessary property for equations complete.