Schur functor

In mathematics, especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the 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 . 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 (1) is multilinear, (2) is alternating in the entries indexed by each column of T, (3) 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 (in order). 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 (3) 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 (left-to-right). Taking (i.e., the numbers in the second column of T) we have while if then 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 = (n) the Schur functor Sλ(V) = Symn(V). For a partition λ = (1, ..., 1) (repeated n times) the Schur functor Sλ(V) = Λn(V).