Rota is not around anymore, so we can't go and ask him what he meant. My guess is that he is referring to the $\lambda$-ring structure of symmetric functions which is related to plethysm and the composition of Schur functors (that's the representation theory connection as well as the category theoretic one regarding polynomial functors). This $\lambda$-ring structure plays a role in $K$-theory as explained, e.g. see "Riemann-Roch Algebra" by Fulton and Lang. Yang's "Lambda-Rings" book and this long survey article about big Witt vectors by Hazewinkel are also nice (and probably more accessible) pedagogical references.
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Corrected the reference to Darij Grinberg's work, as user Darij Grinberg provided two better alternatives to his work in this comment on the answer https://mathoverflow.net/questions/270490/understanding-a-quip-from-gian-carlo-rota#comment668987_270494
Abdelmalek Abdesselam
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