Schur functor

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. 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 ${\displaystyle \mathbb {S} ^{\lambda }V}$ also has a natural action of G for any Schur functor ${\displaystyle \mathbb {S} ^{\lambda }(-)}$.

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 ${\displaystyle \varphi :E^{\times n}\to M}$ satisfying the following conditions

where the sum is over n-tuples x′ obtained from x by exchanging the elements indexed by I with any ${\displaystyle |I|}$ elements indexed by the numbers in column ${\displaystyle i-1}$ (in order).

The universal R-module ${\displaystyle \mathbb {S} ^{\lambda }E}$ that extends ${\displaystyle \varphi }$ to a mapping of R-modules ${\displaystyle {\tilde {\varphi }}:\mathbb {S} ^{\lambda }E\to M}$ is the image of E under the Schur functor indexed by λ.

For an example of the condition (3) placed on ${\displaystyle \varphi }$ suppose that λ is the partition ${\displaystyle (2,2,1)}$ 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 ${\displaystyle I=\{4,5\}}$ (i.e., the numbers in the second column of T) we have

while if ${\displaystyle I=\{5\}}$ then

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

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 S^λ(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 S^λ(V) ⊗ det(V)^⊗m, where λ is a Young diagram with each row strictly shorter than k, and m is any (possibly negative) integer.