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

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?

• Decompositions of the tensor powers of $V$ should be extremely classical; from here the problem reduces to figuring out which irreducible components of the tensor powers appear in these submodules, and shouldn't this reduce to computing the action of these trace and cyclic averaging operations on each irreducible component? (For the cyclic operations this should in turn reduce to Schur-Weyl duality, or something.) – Qiaochu Yuan Jul 2 '14 at 19:31
• One further comment: For the two applications to affine geometry, you really should be using the representation theory of $\mathrm{GL}(n,\mathbb{R})$, not $\mathrm{O}(n)$. For example, your space $\mathscr{C}(V)$ should really be thought of as the subspace of $V\otimes V^\ast\otimes \Lambda^2(V^\ast)$ that is the intersection of the kernels of the two natural maps to $V\otimes \Lambda^3(V^\ast)$ (i.e., skew-symmetrization in the last three indices) and to $\Lambda^2(V^\ast)$ (i.e., contraction in the first two indices). This has only two irreducible $\mathrm{GL}(n,\mathbb{R})$ summands. – Robert Bryant Jul 3 '14 at 12:07

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 $(1,1,0,\ldots,0)$).
• Another reference is Symmetry, Representations, and Invariants by Roe Goodman and Nolan Wallach. They treat decomposition of tensor spaces for $\mathrm{O}(n)$ and $\mathrm{Sp}(n)$ in a freely available appendix: math.rutgers.edu/~goodman/repbook.html – Vít Tuček Jul 8 '14 at 7:46