Schur functor

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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[edit]

If V is a complex vector space of dimension k then either \mathbb{S}^\lambda V is zero, if the 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.

See also[edit]

References[edit]

  • 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[edit]