In mathematics, especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative ring to itself.
Schur functors are indexed by Young diagrams in such a way that the horizontal diagram with n cells corresponds to the nth symmetric power functor, and the vertical diagram with n cells corresponds to the nth exterior power functor.
Schur functors are indexed by partitions and are described as follows.
Let R be a commutative ring, E an R-module and λ a partition of a positive integer n. Let T be a Young tableau of shape λ, thus indexing the factors of the n-fold direct product, E × E × ... × E, with the boxes of T. Consider those maps of R-modules
satisfying the following conditions where the sum is over n-tuples x′ obtained from x by exchanging the elements indexed by I with any
is the image of E under the Schur functor indexed by λ.
and the tableau T is numbered such that its entries are 1, 2, 3, 4, 5 when read top-to-bottom (left-to-right).
The following descriptions hold:[1] Let V be a complex vector space of dimension k. It's a tautological representation of its automorphism group GL(V).
In fact, any rational representation of GL(V) is isomorphic to a direct sum of representations of the form Sλ(V) ⊗ det(V)⊗m, where λ is a Young diagram with each row strictly shorter than k, and m is any (possibly negative) integer.
is the number of standard young tableaux of shape λ.
is the Specht module indexed by λ. Schur functors can also be used to describe the coordinate ring of certain flag varieties.
For two Young diagrams λ and μ consider the composition of the corresponding Schur functors Sλ(Sμ(−)).
From the general theory it is known[1] that, at least for vector spaces over a characteristic zero field, the plethysm is isomorphic to a direct sum of Schur functors.
The problem of determining which Young diagrams occur in that description and how to calculate their multiplicities is open, aside from some special cases like Symm(Sym2(V)).