I know of the following three definitions of the RSK correspondence: (i) Row insertion (or more generally, plactic insertion) (ii) Viennot's construction (iii) Fomin's growth diagrams
However, all of these definitions are rather computational in nature. I wonder if there is a set of natural conditions on a map from permutations to pairs of young tableaux of the same shape which force the map to be RSK?
For example, Green's theorem about the sums $\lambda_1 + \dots + \lambda_k$ equaling the size of the smallest set of letters of the permutation that can be partitioned into k increasing subsequences, where $\lambda$ is the shape of the tableaux, seems to determine the shape but of course not the entries of the tableaux.
A closely related (but subjective) question is: which properties do we (in representation theory) really want from the RSK, and are there other maps which have this property.