Schur Functor
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High Quality Content by WIKIPEDIA articles! 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 a partition of a positive integer n. Let T be a Young tableau of shape , thus indexing the factors of the n-fold direct product, E × E × ... × E, with the boxes of T. Consider those maps of R-modules varphi:E^{times n} to M satisfying the following conditions (1) varphi is…mehr

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High Quality Content by WIKIPEDIA articles! 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 a partition of a positive integer n. Let T be a Young tableau of shape , thus indexing the factors of the n-fold direct product, E × E × ... × E, with the boxes of T. Consider those maps of R-modules varphi:E^{times n} to M satisfying the following conditions (1) varphi is multilinear, (2) varphi is alternating in the entries indexed by each column of T, (3) varphi satisfies an exchange condition stating that if I subset {1,2,dots,n} are numbers from column i of T then varphi(x) = sum_{x'} varphi(x') where the sum is over n-tuples x' obtained from x by exchanging the elements indexed by I with any I elements indexed by the numbers in column i 1 (in order).