I have recently been working on a project that involves tensor calculus. Here I collect some notions and identities, to spare myself from googling them each time.

We work on quasicoherent sheaves on a scheme, and when no ambiguity arises we omit this from the notation.

Schur functors

Here there will be an explanation of what Schur functors are and of how they generalize the symmetric and exterior products.

For the moment one can read chapter $6$ of [2].

Identities

\[\DeclareMathOperator{\Sym}{Sym} \begin{align} \bigwedge^{n}\left(A \oplus B\right) ~~&\simeq~~ \bigoplus_{p + q = n} \bigwedge^{p}(A) \otimes \bigwedge^{q}(B) \\ \bigwedge^{n}\left(A \otimes B\right) ~~&\simeq~~ \bigoplus_{|a| = n} \Sigma_{a} (A) \otimes \Sigma_{a^{*}} (B) \\ \Sym^{n}\left(A \oplus B\right) ~~&\simeq~~ \bigoplus_{p + q = n} \Sym^{p}(A) \otimes \Sym^{q}(B) \\ \Sym^{n}\left(A \otimes B\right) ~~&\simeq~~ \bigoplus_{|a| = n} \Sigma_{a} (A) \otimes \Sigma_{a} (B) \end{align}\]

Proof. The identities splitting direct sums are an immediate corollary of [Exercise II.5.16, 1].

The third item of the exercise states that for any short exact sequence \(0 \to A \to B \to C \to 0,\) there are filtrations

\[0 = F_{n+1} \subset F_{n} \subset \dots \subset F_{1} \subset F_{0} = \Sym^{n}(B)\]

and

\[0 = G_{n+1} \subset G_{n} \subset \dots \subset G_{1} \subset G_{0} = \bigwedge^{n}(B),\]

such that

\[F_{i}/F_{i+1} \simeq \Sym^{i}(A) \otimes \Sym^{n-i}(C)\]

and

\[G_{i}/G_{i+1} \simeq \bigwedge^{i}(A) \otimes \bigwedge^{n-i}(C).\]

In the cases of two summands the proof is immediate, and the general cases follow by induction.

The identities splitting tensor products are proved in [Exercise 6.11, 2].

$\square$

References

[1] Algebraic Geometry, R. Hartshorne.

[2] Representation Theory: A First Course, W. Fulton, J. Harris.