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 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

• Young symmetrizer
• Schur polynomial

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