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 "n"th exterior power functor, and the vertical diagram with "n" cells corresponds to the "n"th symmetric power functor. If a vector space "V" is a representation of a group "G", then formula_1 also has a natural action of "G" for any Schur functor formula_2.
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, "E" × "E" × ... × "E", with the boxes of "T". Consider those maps of "R"-modules formula_3 satisfying the following conditions
(1) formula_4 is multilinear,
(2) formula_4 is alternating in the entries indexed by each column of "T",
(3) formula_4 satisfies an exchange condition stating that if formula_7 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 formula_9 elements indexed by the numbers in column formula_10 (in order).
The universal "R"-module formula_11 that extends formula_4 to a mapping of "R"-modules formula_13 is the image of "E" under the Schur functor indexed by λ.
For an example of the condition (3) placed on formula_4
suppose that λ is the partition formula_15 and the tableau
"T" is numbered such that its entries are 1, 2, 3, 4, 5 when read
top-to-bottom (left-to-right). Taking formula_16 (i.e.,
the numbers in the second column of "T") we have
while if formula_18 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:
Applications.
Let "V" be a complex vector space of dimension "k". It's a tautological representation of its automorphism group GL("V"). If λ is a diagram where each row has no more than "k" cells, then Sλ("V") is an irreducible GL("V")-representation of highest weight λ. 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.
In this context Schur-Weyl duality states that as a formula_21-module
where formula_23 is the number of standard young tableaux of shape λ. More generally, we have the decomposition of the tensor product as formula_24-bimodule
where formula_26 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 Sλ(Sμ(-)). This composition is called a plethysm of λ and μ. From the general theory it's 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"(Sym2("V")).