Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry

Question

Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a $\mathrm{O}(n)$-module via the diagonal action. I am interested in the following specific submodules, which arise from Riemannian geometry and affine differential geometry:

(1) $ \mathscr{R}(V)=\{R\in V^{\otimes 4}\mid\mathrm{Cyc}_{12}(R)=\mathrm{Cyc}_{34}(R)=\mathrm{Cyc}_{234}(R)=0\} $. Here $\mathrm{Cyc}_{234}: V^{\otimes4}\rightarrow V^{\otimes 4} $ is the $\mathrm{O}(n)$-module morphism which averages cyclic permutations of the 2nd, 3rd and 4th components, $$ \mathrm{Cyc}_{234}(x_1\otimes x_2\otimes x_3\otimes x_4)=\frac{1}{3}\left(x_1\otimes x_2\otimes x_3\otimes x_4+x_1\otimes x_3\otimes x_4\otimes x_2+x_1\otimes x_4\otimes x_2\otimes x_3\right). $$ $\mathrm{Cyc}_{12}$ is defined similarly. $\mathscr{R}(V)$ is the space where Riemannian curvature tensors live.

(2) $ \mathscr{C}(V)=\{R\in V^{\otimes 4}\mid \mathrm{Tr}_{12}(R)=0,\, \mathrm{Cyc}_{34}(R)=\mathrm{Cyc}_{234}(R)=0\}. $ Here $\mathrm{Tr}_{12}: V^{\otimes 4}\rightarrow V\otimes V$ contracts the 1st and 2nd components through $g$. $\mathscr{C}(V)$ is the space where curvature tensors of torsion-free volume-preserving affine connections on a Riemannian manifold live.

(3) $ \mathscr{A}(V)=\{A\in V^{\otimes3}\mid\mathrm{Cyc}_{12}(A)=A, \, \rm{Tr}_{23}(A)=0\} $, the space where the difference tensor of two torsion-free volume-preserving affine connections lives.

The decomposition of $\mathscr{R}(V)$ into irreducible $\mathrm{O}(n)$-modules is well known under the name "Ricci decomposition" and is discussed briefly in the book "Einstein Manifolds" by Besse. The decomposition of $\mathscr{C}(V)$ seems obtained only recently by Blazic, Gilkey, Nikcevic, and Simon. I can't find any reference for the decomposition of $\mathscr{A}(V)$. So my question is

(1) Where exactly can I find a systematic treatment of the decomposition problem for this kind of $\mathrm{O}(n)$-modules?

(2) In particular, given a $\mathrm{O}(n)$-module $W$ like the above ones, i.e. a submodule of $V^{\otimes k}$ which is the kernel of some $\mathrm{Tr}$ and $\mathrm{Cyc}$ type morphisms, is there a general algorithm, or strategy, to decompose $W$ into irreducible pieces?

(3) In particular, has a irreducible decomposition of $\mathscr{A}(V)$ appeared in the literature?

Answer 1

As Qiaochu Yuan wrote, these irreducible decompositions are classical. You can read about them in Hermann Weyl's The Classical Groups (for example), and the questions you are asking are basically exercises in writing out what the standard methods tell you. (Of course, there are more recent treatments, which modern readers might find more readable than Weyl. You might, for example, read about the method of Young tableau in the popular recent book Representation Theory by Fulton and Harris, which seems to be what you need.)

For example, Weyl's formula gives you that $\mathscr{A}(V)$ is the direct sum of two irreducible subspaces; one of them, in your notation, is the subspace $$ \mathscr{A}_1(V)=\{A\in V^{\otimes3}\mid\mathrm{Cyc}_{12}(A)=A,\,\mathrm{Cyc}_{23}(A)=A,\,\mathrm{Tr}_{23}(A)=0\}, $$ but it is better known as the space of harmonic (i.e., traceless) symmetric cubic polynomials on $V$ and has dimension $\tfrac16 n(n{-}1)(n{+}4)$, while the other is $$ \mathscr{A}_2(V)=\{A\in V^{\otimes3}\mid\mathrm{Cyc}_{12}(A)=A,\,\mathrm{Cyc}_{123}(A)=0,\,\mathrm{Tr}_{23}(A)=0\} $$ and has dimension $\tfrac16 n(n{+}1)(2n{+}1)$ (it is usually described as the $\mathrm{O}(n)$-representation with highest weight vector $(1,1,0,\ldots,0)$).