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 $\rho$, denoted ${\mathbb{S}}_{\rho }$, 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 $K{S}_n$. Now, the functor is defined as follows:

• On objects: It sends a vector space $V$ to the tensor product of modules $V^{\otimes n}{\otimes }_{K{S}_n}W$. 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 $K{S}_n$-module where $S_n$ acts by permuting the tensor product factors. We now take its tensor product as a $K{S}_n$-module with $W$. This is not the same as taking the tensor product of vector spaces. Note that since $K{S}_n$ is not commutative, we have to treat $W$ as a left $K{S}_n$-module and $V^{\otimes n}$ as a right $K{S}_n$-module. Also note that once we have taken the tensor product, we no longer have a $K{S}_n$-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 $K{S}_n$ 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.

Combinatorial definition for finite-dimensional vector space in terms of Young tableaux

The Schur functor corresponding to a partition $\lambda$ of $n$ applied to the vector space $K^m$ (with $m$ unrelated to $n$) can be defined combinatorially as follows. As a vector space, it has basis the set of semistandard Young tableaux of shape $\lambda$ and all entries between 1 and $m$.

Definition as a functor from representations to representations

This definition can be obtained using abstract nonsense from the definition for vector spaces. For a representation $\alpha :G\to GL(V)$ and a linear representation $\rho$ of a symmetric group $S_n$, we define ${\mathbb{S}}_{\rho }(\alpha )$ as a representation of $G$ on the vector space $S_{\rho }(V)$, with the map:

$G\to {\mathbb{S}}_{\rho }(V)$

given by:

$g\mapsto {\mathbb{S}}_{\rho }(\alpha (g))$

where the latter ${\mathbb{S}}_{\rho }$ denotes the effect of the functor on morphisms.

Facts

• Formula for calculating effect of Schur functor on character
• We have the following:

$V^{\otimes n}={⨁}_{\lambda }{\mathbb{S}}_{\lambda }^{d_{\lambda }}$

where $d_{\lambda }$ denotes the degree of the irreducible representation of $S_n$ corresponding to $\lambda$. The notation here means that ${\mathbb{S}}_{\lambda }$ is added to itself $d_{\lambda }$ times.

Note that although ${\mathbb{S}}_{\lambda }$ does not have a natural $S_n$-action, ${\mathbb{S}}_{\lambda }^{d_{\lambda }}$ does.