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We propose a way to separate variables in a rational integrable$\mathfrak{gl}(n)$ spin chain with an arbitrary finite-dimensional irreduciblerepresentation at each site and with generic twisted periodic boundaryconditions. Firstly, we construct a basis that diagonalises a higher-rankversion of the Sklyanin B-operator; the construction is based on recursiveusage of an embedding of a $\mathfrak{gl}(k)$ spin chain into a$\mathfrak{gl}(k+1)$ spin chain which is induced from a Yangian homomorphismand controlled by dual diagonals of Gelfand-Tsetlin patterns. Then, we showthat the same basis can be equivalently constructed by action ofBacklund-transformed fused transfer matricies, whence the Bethe wave functionsfactorise into a product of ascending Slater determinants in BaxterQ-functions. Finally, we construct raising and lowering operators -- theconjugate momenta -- as normal-ordered Wronskian expressions in BaxterQ-operators evaluated at zeros of B -- the separated variables. It is animmediate consequence of the proposed construction that the Bethe algebracomprises the maximal possible number of mutually commuting charges -- anecessary property for Bethe equations to be complete.

Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow