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I am afraid questions might be trivial and well-known, well, nevervethless. I'll ask it.

Consider category of vector spaces. There are naturally defined functors to itself : take vector space and consider its symmetric or antisymmetric powers, more general things can be constructed - which is called Schur functors (as far as I understand) they are indexed by Young diagrams. (They are defined as certain subspaces in tensor product $V^{\otimes n}$, for example if we take $V^{\otimes 3}$ then there are symmetric tensors, antisymmetric and there is certain complement to them - that is one of the Schur functors).

So we get many functors $F_{diagr}$. Functors can be composed. Are there any relations between composition of functors ? E.g. $F_{d1} F_{d2} = F_{d3} F_{d4}$ or whatever ?

Also functors on the catetory of vectors spaces can be added - so we have some algebra. What are the relations in this algebra ?

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You are asking about plethysms. – Wilberd van der Kallen Feb 20 '12 at 9:26
Wilberd van der Kellen's is right, but there might be some more to say here. The perspective in the plethysm literature is usually to write a composition of two Schur functors as a linear combination of other Schur functors -- like describing a ring by giving a multiplication table for basis elements. It might be interesting to think about describing this ring by generators and relations instead. – David Speyer Feb 20 '12 at 17:55
Some plethysym relations in special cases are known, and every few years a new paper comes out with another special case. This is known to be a hard problem in general. – Alexander Woo Feb 20 '12 at 17:58

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