  The (dual) Cauchy identity has an easy algebraic proof utilising a
commutation relation between the up and (dual) down operators. By using Fomin's
growth diagrams, a bijective proof of the commutation relation can be
"bijectivised" to obtain RSK like correspondences. In this paper we give a
concise overview of this machinery and extend it to Littlewood type identities
by introducing a new family of relations between these operators, called
projection identities. Thereby we obtain infinite families of bijections for
the Littlewood identities generalising the classical ones. We believe that this
approach will be useful for finding bijective proofs for Littlewood type
identities in other settings such as for Macdonald polynomials and their
specialisations, alternating sign matrices or vertex models.
