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.

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

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Schur functor

Applications

See also

References

External links