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In mathematics, especially in the field of representation theory, a Schur functor is a functor from the category of modules over a fixed commutative ring to itself. 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
If V is a complex vector space of dimension k then either is zero, if the length of λ is longer than k, or it is an irreducible representation of highest weight λ.
In this context Schur-Weyl duality states that as a -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.
- J. Towber, Two new functors from modules to algebras, J. Algebra 47 (1977), 80-104. doi:10.1016/0021-8693(77)90211-3
- W. Fulton, Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997, ISBN 0-521-56724-6.