Symmetries of Tensors: from manipulation of indices to group representation theory

This piece is a set of supplementary notes on General Relativity, and it may also serve as a extended reading for tensor analysis.

This note extends the familiar rank-2 split,
$$
T_{ab}=T_{(ab)}+T_{[ab]},
$$
to rank-$n$ tensors via the natural action of the symmetric group $S_n$ that permutes tensor factors of $V^{\otimes n}$.
Invariant subspaces ↔ irreps of $S_n$.
Symmetry types of indices correspond to irreducible $S_n$-modules (Specht modules) labeled by Young diagrams $\lambda\vdash n$.
Projectors.
Central (character) projectors select the $\lambda$-isotypic component; Young symmetrizers further isolate a single copy with the prescribed row-symmetry / column-antisymmetry.
Schur–Weyl viewpoint.
Commuting actions of $S_n$ and $\mathrm{GL}(V)$ yield
$$
V^{\otimes n}\cong\bigoplus_{\lambda\vdash n,\ \ell(\lambda)\le \dim V} S_\lambda(V)\otimes \mathrm{Specht}_\lambda,
$$
explaining multiplicities and the viability condition $\ell(\lambda)\le \dim V$.
How to use: pick a Young diagram $\lambda$, apply the one-line character projector to any $T\in V^{\otimes n}$ to get its $\lambda$-component, and optionally use a Young symmetrizer to extract one mixed-symmetry copy. The rank-2 case appears as $\lambda=(2)$ (totally symmetric) and $\lambda=(1,1)$ (totally antisymmetric).