Schur functor
Encyclopedia
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
satisfying the following conditions
(1)
is multilinear,
(2)
is alternating in the entries indexed by each column of T,
(3)
satisfies an exchange condition stating that if 
are numbers from column i of T then
where the sum is over n-tuples x' obtained from x by exchanging the elements indexed by I with any
elements indexed by the numbers in column 
(in order).
The universal R-module
that extends 
to a mapping of R-modules 
is the image of E under the Schur functor indexed by λ.
For an example of the condition (3) placed on
suppose that λ is the partition
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.e.,
the numbers in the second column of T) we have
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the field of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, a Schur functor is a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over a fixed commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
to itself. Schur functors are indexed by partitions
Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
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
Young tableau
In mathematics, a Young tableau is a combinatorial object useful in representation theory. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at...
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
satisfying the following conditions(1)
is multilinear,(2)
is alternating in the entries indexed by each column of T,(3)
satisfies an exchange condition stating that if are numbers from column i of T then
where the sum is over n-tuples x' obtained from x by exchanging the elements indexed by I with any
elements indexed by the numbers in column (in order).The universal R-module
that extends to a mapping of R-modules is the image of E under the Schur functor indexed by λ.For an example of the condition (3) placed on
suppose that λ is the partition
and the tableauT is numbered such that its entries are 1, 2, 3, 4, 5 when read
top-to-bottom, left-to-right). Taking
(i.e.,the numbers in the second column of T) we have
-
while if
then
-
Applications
If V is a complex vector space of dimension k then either
is zero, if then length of λ is longer
than k, or it is an irreducible
representation of
highest weight λ.
In this context Schur-Weyl duality states that as a
-module
where
is the number of standard young tableaux of shape λ. More generally, we have the decomposition of the tensor product as 
-bimodule
where
is the Specht moduleSpecht moduleIn mathematics, a Specht module is one of the representations of symmetric groups studied by .They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of n form a complete set of irreducible representations of the symmetric group on n points.-Definition:Fix a...
indexed by λ. Schur functors can also be used to describe the coordinate ring of certain flag varieties.
External links
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