Schur functor

In mathematics, especially in the field of representation theory, Schur functors are certain functors from the category of modules over a fixed commutative ring to itself. They generalize the constructions of exterior powers and symmetric powers of a vector space. 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. If a vector space V is a representation of a group G, then also has a natural action of G for any Schur functor.

Definition

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,, with the boxes of T. Consider those maps of R-modules satisfying the following conditions

is multilinear,
is alternating in the entries indexed by each column of T,
satisfies an exchange condition stating that if are numbers from column i of T then

where the sum is over n-tuples x′ obtained from x by exchanging the elements indexed by I with any elements indexed by the numbers in column .
The universal R-module that extends to a mapping of R-modules is the image of E under the Schur functor indexed by λ.
For an example of the condition placed on
suppose that λ is the partition and the tableau
T is numbered such that its entries are 1, 2, 3, 4, 5 when read
top-to-bottom. Taking we have
while if then

Examples

Fix a vector space V over a field of characteristic zero. We identify partitions and the corresponding Young diagrams. The following descriptions hold:

For a partition λ = the Schur functor S^λ = Symⁿ.
For a partition λ = the Schur functor S^λ = Λⁿ.
For a partition λ = the Schur functor S^λ is the cokernel of the comultiplication map of exterior powers Λ³ → Λ² ⊗ V.
For a partition λ = the Schur functor S^λ is the quotient of Λ² ⊗ Λ² by the images of two maps. One is the composition Λ³ ⊗ V → Λ² ⊗ V ⊗ V → Λ² ⊗ Λ², where the first map is the comultiplication along the first coordinate. The other map is a comultiplication Λ⁴ → Λ² ⊗ Λ².
For a partition λ =, with 1 repeated m times, the Schur functor S^λ is the quotient of Λⁿ ⊗ Sym^m by the image of the composition of the comultiplication in exterior powers and the multiplication in symmetric powers:
:

Applications

Let V be a complex vector space of dimension k. It's a tautological representation of its automorphism group GL. If λ is a diagram where each row has no more than k cells, then S^λ is an irreducible GL-representation of highest weight λ. In fact, any rational representation of GL is isomorphic to a direct sum of representations of the form S^λ ⊗ det^⊗m, where λ is a Young diagram with each row strictly shorter than k, and m is any integer.
In this context Schur-Weyl duality states that as a GL-module
where is the number of standard young tableaux of shape λ. More generally, we have the decomposition of the tensor product as -bimodule
where is the Specht module indexed by λ. Schur functors can also be used to describe the coordinate ring of certain flag varieties.

Plethysm

For two Young diagrams λ and μ consider the composition of the corresponding Schur functors S^λ. This composition is called a plethysm of λ and μ. From the general theory it is known 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 Sym^m.