The composite of Schur functors is again a Schur functor. The process of composing Schur functors is known as plethysm, especially in situations where we want an explicit “formula” for how the Schur functors compose.
as a direct sum of Schur functors coming from Young diagrams. This is the sort of problem people study when they talk about “plethysm”.
Let be the skeletal category of finite sets and bijections and a symmetric additive monoidal category with monoidal product and unit object . Objects in the category of contravariant functors can be descibed as collections of objects in with action of a symmetric group on letters. The category acts on by the polynomial functors
The composition of such functors defines a monoidal product on called the plethysm product. This way we obtain a symmetric monoidal category. The monoids in that category are the (symmetric) -operads.
In Richard P. Stanley’s book Enumerative Combinatorics, he discusses the origin of the term ‘plethysm’ in Volume 2, Appendix 2. He says that the term was introduced in
The term ‘plethysm’ was suggested to Littlewood by M. L. Clark after the Greek word plethysmos, or πληθυσμός, which means ‘multiplication’ in modern Greek (though apparently the meaning goes back to ancient Greek). The related term plethys in Greek means ‘a big number’ or ‘a throng’, and this in turn comes from the Greek verb plethein, which means ‘to be full’, ‘to increase’, ‘to fill’, etc.
Some aspects of plethysm appear (partly through exercises) in the textbook W. Fulton, J. Harris, Representation theory.
For an application in the study of characteristic classes see
An application in a counting problem: