# Schur functor

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 $\varphi:E^{\times n} \to M$ 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).

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

$\varphi(x_1,x_2,x_3,x_4,x_5) = \varphi(x_5,x_2,x_3,x_4,x_1) + \varphi(x_1,x_5,x_3,x_4,x_2) + \varphi(x_1,x_2,x_5,x_4,x_3).$

## Applications

If V is a complex vector space of dimension k then either $\mathbb{S}^\lambda V$ is zero, if then length of λ is longer than k, or it is an irreducible $GL(V)$ representation of highest weight λ.

In this context Schur-Weyl duality states that as a $GL(V)$-module

$V^{\otimes n} = \bigoplus_{\lambda \vdash n: \ell(\lambda) \leq k} (\mathbb{S}^{\lambda} V)^{\oplus f^\lambda}$

where $f^\lambda$ is the number of standard young tableaux of shape λ. More generally, we have the decomposition of the tensor product as $GL(V) \times \mathfrak{S}_n$-bimodule

$V^{\otimes n} = \bigoplus_{\lambda \vdash n: \ell(\lambda) \leq k} (\mathbb{S}^{\lambda} V) \otimes \operatorname{Specht}(\lambda)$

where $\operatorname{Specht}(\lambda)$ is the Specht module indexed by λ. Schur functors can also be used to describe the coordinate ring of certain flag varieties.

## References

• J. Towber, Two new functors from modules to algebras, J. Algebra 47 (1977), 80-104.
• W. Fulton, Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997, ISBN 0-521-56724-6.