Schur functor

Definition as a functor from vector spaces to vector spaces

Schur functor associated with a representation of the symmetric group

Suppose $\rho :S_{n}\to GL(W)$ is a linear representation of the symmetric group $S_{n}$ over a field $K$. The Schur functor associated with $\rho$ is a functor from the category of $K$-vector spaces to the category of $K$-vector spaces defined as follows. First, using the equivalence of definitions of linear representation, consider $W$ as a module over the group ring $KS_{n}$. Now, the functor is defined as follows:

• On objects: It sends a vector space $V$ to the tensor product of modules $V^{\otimes n}\otimes _{KS_{n}}W$. Here is what this means. $V^{\otimes n}$ is the $n$-fold tensor product of $V$ as a $K$-vector space. This naturally acquires the structure of a $KS_{n}$-module where $S_{n}$ acts by permuting the tensor product factors. We now take its tensor product as a $KS_{n}$-module with $W$. This is not the same as taking the tensor product of vector spaces. Note that since $KS_{n}$ is not commutative, we have to treat $W$ as a left $KS_{n}$-module and $V^{\otimes n}$ as a right $KS_{n}$-module. Also note that once we have taken the tensor product, we no longer have a $KS_{n}$-module structure, just a $K$-vector space.
• On morphisms: Given a linear map $\mu :V_{1}\to V_{2}$, it induces a map $\mu ^{\otimes n}:V_{1}^{\otimes n}\to V_{2}^{\otimes n}$ which in turn induces a map of the tensor products with $W$. Functoriality must be checked, but is true.

Schur functor associated with a partition

Assume that $K$ has characteristic zero. Suppose $\lambda$ is a unordered integer partition of a positive integer $n$. The Schur functor associated with $\lambda$, denoted $\mathbb {S} _{\lambda }$, is defined as the Schur functor associated with the irreducible linear representation of $S_{n}$ corresponding to $\lambda$ (see linear representation theory of symmetric groups).

In this case, the module for the irreducible representation can be thought of explicitly as a left ideal inside $KS_{n}$ and the tensor product operation can be thought of as multiplication on the right by this ideal. Note that multiplication by the two-sided ideal corresponding to the representation would give a sum of the Schur functor with itslf degree of the representation many times.

The result also holds if the characteristic of $K$ is greater than $n$. If $K$ has characteristic a prime less than or equal to $n$, there is some ambiguity about how to define the Schur functor.

Definition as a functor from representations to representations

This definition can be obtained using abstract nonsense from the definition for vector spaces. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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Facts

• Formula for calculating effect of Schur functor on character