Schur functor
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In mathematics, especially in the field of
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 studies modules over these abstract algebraic structures. In essen ...
, Schur functors (named after
Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at th ...
) are certain
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s from the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a fixed commutative ring to itself. They generalize the constructions of
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
s and
symmetric power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n. More precisely, the notion exists at least in the ...
s of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. Schur functors are indexed by
Young diagram In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups ...
s in such a way that the horizontal diagram with ''n'' cells corresponds to the ''n''th symmetric power functor, and the vertical diagram with ''n'' cells corresponds to the ''n''th exterior power functor. If a vector space ''V'' is a representation of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
''G'', then \mathbb^V also has a natural action of ''G'' for any Schur functor \mathbb^(-).


Definition

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 In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups a ...
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^ \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 \ are numbers from column ''i'' of ''T'' then : \varphi(x) = \sum_ \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^\lambda E that extends \varphi to a mapping of ''R''-modules \tilde:\mathbb^\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 = \ (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 = \ 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

Fix a vector space ''V'' over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of characteristic zero. We identify partitions and the corresponding Young diagrams. The following descriptions hold: * For a partition λ = (n) the Schur functor ''S''λ(''V'') = Sym''n''(''V''). * For a partition λ = (1, ..., 1) (repeated ''n'' times) the Schur functor ''S''λ(''V'') = Λ''n''(''V''). * For a partition λ = (2, 1) the Schur functor ''S''λ(''V'') is the
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
of the
comultiplication In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
map of exterior powers Λ3(''V'') → Λ2(''V'') ⊗ ''V''. * For a partition λ = (2, 2) the Schur functor ''S''λ(''V'') is the quotient of Λ2(''V'') ⊗ Λ2(''V'') by the images of two maps. One is the composition Λ3(''V'') ⊗ ''V'' → Λ2(''V'') ⊗ ''V'' ⊗ ''V'' → Λ2(''V'') ⊗ Λ2(''V''), where the first map is the comultiplication along the first coordinate. The other map is a comultiplication Λ4(''V'') → Λ2(''V'') ⊗ Λ2(''V''). * For a partition λ = (''n'', 1, ..., 1), with 1 repeated ''m'' times, the Schur functor ''S''λ(''V'') is the quotient of Λ''n''(''V'') ⊗ Sym''m''(''V'') by the image of the composition of the comultiplication in exterior powers and the multiplication in symmetric powers: : \Lambda^(V) \otimes \mathrm^(V) \xrightarrow \Lambda^n(V) \otimes V \otimes \mathrm^(V) \xrightarrow \Lambda^n(V) \otimes \mathrm^m(V)


Applications

Let ''V'' be a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
vector space of dimension ''k''. It's a tautological representation of its
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
GL(''V''). If λ is a diagram where each row has no more than ''k'' cells, then Sλ(''V'') is an irreducible GL(''V'')-representation of highest weight λ. In fact, any rational representation of GL(''V'') is isomorphic to a direct sum of representations of the form Sλ(''V'') ⊗ det(''V'')⊗''m'', where λ is a Young diagram with each row strictly shorter than ''k'', and ''m'' is any (possibly negative) integer. In this context Schur-Weyl duality states that as a GL(V)-module : V^ = \bigoplus_ (\mathbb^ V)^ 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_n-bimodule : V^ = \bigoplus_ (\mathbb^ V) \otimes \operatorname(\lambda) where \operatorname(\lambda) is the Specht module indexed by λ. Schur functors can also be used to describe the coordinate ring of certain flag varieties.


Plethysm

For two Young diagrams λ and μ consider the composition of the corresponding Schur functors Sλ(Sμ(-)). This composition is called a plethysm of λ and μ. From the general theory it's known that, at least for vector spaces over a characteristic zero field, the plethysm is isomorphic to a direct sum of Schur functors. The problem of determining which Young diagrams occur in that description and how to calculate their multiplicities is open, aside from some special cases like Sym''m''(Sym2(''V'')).


See also

* Young symmetrizer *
Schur polynomial In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In ...
*
Littlewood–Richardson rule In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural number ...
*
Polynomial functor In algebra, a polynomial functor is an endofunctor on the category \mathcal of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers V \mapsto \operatorname^n(V) and the exterior powers V \ ...


References

* J. Towber, Two new functors from modules to algebras, J. Algebra 47 (1977), 80-104. doi:10.1016/0021-8693(77)90211-3 * W. Fulton, ''Young Tableaux, with Applications to Representation Theory and Geometry''. Cambridge University Press, 1997, {{isbn, 0-521-56724-6. *


External links


Schur Functors , The n-Category Café
Representation theory Functors