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
φ
:
E
×
n
→
M
satisfying the following conditions
(1)
φ
is multilinear,
(2)
φ
is alternating in the entries indexed by each column of T,
(3)
φ
satisfies an exchange condition stating that if
I
⊂
{
1
,
2
,
…
,
n
}
are numbers from column i of T then
φ
(
x
)
=
∑
x
′
φ
(
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
S
λ
E
that extends
φ
to a mapping of R-modules
φ
~
:
S
λ
E
→
M
is the image of E under the Schur functor indexed by λ.
For an example of the condition (3) placed on
φ
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
φ
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
=
φ
(
x
4
,
x
5
,
x
3
,
x
1
,
x
2
)
+
φ
(
x
4
,
x
2
,
x
5
,
x
1
,
x
3
)
+
φ
(
x
1
,
x
4
,
x
5
,
x
2
,
x
3
)
,
while if
I
=
{
5
}
then
φ
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
=
φ
(
x
5
,
x
2
,
x
3
,
x
4
,
x
1
)
+
φ
(
x
1
,
x
5
,
x
3
,
x
4
,
x
2
)
+
φ
(
x
1
,
x
2
,
x
5
,
x
4
,
x
3
)
.
If V is a complex vector space of dimension k then either
S
λ
V
is zero, if the length of λ is longer than k, or it is an irreducible
G
L
(
V
)
representation of highest weight λ.
In this context Schur-Weyl duality states that as a
G
L
(
V
)
-module
V
⊗
n
=
⨁
λ
⊢
n
:
ℓ
(
λ
)
≤
k
(
S
λ
V
)
⊕
f
λ
where
f
λ
is the number of standard young tableaux of shape λ. More generally, we have the decomposition of the tensor product as
G
L
(
V
)
×
S
n
-bimodule
V
⊗
n
=
⨁
λ
⊢
n
:
ℓ
(
λ
)
≤
k
(
S
λ
V
)
⊗
Specht
(
λ
)
where
Specht
(
λ
)
is the Specht module indexed by λ. Schur functors can also be used to describe the coordinate ring of certain flag varieties.
