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, , with the boxes of
T. Consider those maps of
R-modules \varphi:E^{\times n} \to M satisfying the following conditions • \varphi is multilinear, • \varphi is alternating in the entries indexed by each column of
T, • \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). The universal
R-module \mathbb{S}^\lambda E that extends \varphi to a mapping of
R-modules \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 \varphi suppose that
λ is the partition (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 I = \{4,5\} (i.e., the numbers in the second column of
T) we have : \varphi(x_1,x_2,x_3,x_4,x_5) = \varphi(x_4,x_5,x_3,x_1,x_2) + \varphi(x_4,x_2,x_5,x_1,x_3) + \varphi(x_1,x_4,x_5,x_2,x_3), while if I = \{5\} then : \varphi(x_1,x_2,x_3,x_4,x_5) = \varphi(x_5,x_2,x_3,x_4,x_1) + \varphi(x_1,x_5,x_3,x_4,x_2) + \varphi(x_1,x_2,x_5,x_4,x_3). == Examples ==