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| {{Refimprove|date=August 2009}}
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| In [[mathematics]], especially in the field of [[representation theory]], a '''Schur functor''' is a [[functor]] from the [[category (mathematics)|category]] of [[module (mathematics)|modules]] over a fixed [[commutative ring]] to itself. Schur functors are indexed by [[partition (number theory)|partitions]] and are described as follows. Let ''R'' be a commutative ring, ''E'' an ''R''-module
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| 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 of modules|direct product]], ''E'' × ''E'' × ... × ''E'', with the boxes of ''T''. Consider those maps of ''R''-modules <math>\varphi:E^{\times
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| n} \to M</math> satisfying the following conditions
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| (1) <math>\varphi</math> is multilinear,
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| (2) <math>\varphi</math> is alternating in the entries indexed by each column of ''T'',
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| (3) <math>\varphi</math> satisfies an exchange condition stating that if <math>I \subset
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| \{1,2,\dots,n\}</math> are numbers from column ''i'' of ''T'' then
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| : <math>\varphi(x) = \sum_{x'} \varphi(x') </math>
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| where the sum is over ''n''-tuples ''x' '' obtained from ''x'' by exchanging the elements indexed by ''I'' with any <math>|I|</math> elements indexed by the numbers in column <math>i-1</math> (in order).
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| The universal ''R''-module <math>\mathbb{S}^\lambda E</math> that extends <math>\varphi</math> to a mapping of ''R''-modules <math>\tilde{\varphi}:\mathbb{S}^\lambda E \to M</math> is the image of ''E'' under the Schur functor indexed by λ.
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| For an example of the condition (3) placed on <math>\varphi</math>
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| suppose that λ is the partition <math>(2,2,1)</math> and the tableau
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| ''T'' is numbered such that its entries are 1, 2, 3, 4, 5 when read
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| top-to-bottom, left-to-right). Taking <math>I = \{4,5\}</math> (i.e.,
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| the numbers in the second column of ''T'') we have
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| : <math>\varphi(x_1,x_2,x_3,x_4,x_5) =
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| \varphi(x_4,x_5,x_3,x_1,x_2) +
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| \varphi(x_4,x_2,x_5,x_1,x_3) +
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| \varphi(x_1,x_4,x_5,x_2,x_3),</math>
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| while if <math>I = \{5\}</math> then
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| : <math>\varphi(x_1,x_2,x_3,x_4,x_5) = | |
| \varphi(x_5,x_2,x_3,x_4,x_1) +
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| \varphi(x_1,x_5,x_3,x_4,x_2) +
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| \varphi(x_1,x_2,x_5,x_4,x_3).</math>
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| == Applications ==
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| If ''V'' is a complex vector space of dimension ''k'' then either
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| <math>\mathbb{S}^\lambda V</math> is zero, if then length of λ is longer
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| than ''k'', or it is an irreducible <math>GL(V)</math> representation of
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| highest weight λ.
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| In this context [[Schur-Weyl duality]] states that as a <math>GL(V)</math>-module
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| : <math>V^{\otimes n} = \bigoplus_{\lambda \vdash n: \ell(\lambda) \leq k} (\mathbb{S}^{\lambda} V)^{\oplus f^\lambda}</math> | |
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| where <math>f^\lambda</math> is the number of standard young tableaux of shape λ. More generally, we have the decomposition of the tensor product as <math>GL(V) \times \mathfrak{S}_n</math>-bimodule
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| : <math>V^{\otimes n} = \bigoplus_{\lambda \vdash n: \ell(\lambda) \leq k} (\mathbb{S}^{\lambda} V) \otimes \operatorname{Specht}(\lambda)</math>
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| where <math>\operatorname{Specht}(\lambda)</math> is the [[Specht module]] indexed by λ. Schur functors can also be used to describe the coordinate ring of certain flag varieties.
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| ==See also==
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| *[[Young symmetrizer]]
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| == References ==
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| * J. Towber, Two new functors from modules to algebras, J. Algebra 47 (1977), 80-104.
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| * W. Fulton, ''Young Tableaux, with Applications to Representation Theory and Geometry''. Cambridge University Press, 1997, ISBN 0-521-56724-6.
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| == External links ==
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| * [http://golem.ph.utexas.edu/category/2007/04/schur_functors.html Schur Functors | The n-Category Café]
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| [[Category:Representation theory]]
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