# Studying representations of orthogonal group via symmetric group?

Let $$V$$ be the vector space of $$n\times n$$ symmetric matrices with real entries. On $$V$$ we have the natural action of the orthogonal group $$\textrm{O}(n)$$ defined by $$g.A:=g\cdot A\cdot g^{t}$$ for $$g\in\textrm{O}(n)$$ and $$A\in V$$. We identify $$\mathbb{R}^n$$ with the set of diagonal matrices. Thus we get an inclusion $$\textrm{diag}:\mathbb{R}^n\hookrightarrow V$$. Now let $$G=\mathfrak{S}_n$$ be the symmetric group on $$n$$ elements. Considering $$G$$ as the subgroup of $$\textrm{O}(n)$$ consisting of permutation matrices, we get an action of $$G$$ both on $$\mathbb{R}^n$$ and $$V$$ that makes $$\textrm{diag}$$ to a homomorphism of $$G$$-modules. If we pass to the dual map, we get a surjective homomorphism of $$G$$-modules $$\textrm{diag}^\vee:V\to\mathbb{R}^n.$$ The decomposition of the $$\textrm{O}(n)$$-module $$V$$ into irreducibles is $$(\mathbb{R}\cdot\textrm{I}_n)\oplus V_0$$ where $$\textrm{I}_n$$ is the identity matrix and $$V_0$$ is the set of traceless matrices. We note that the image under $$\textrm{diag}^\vee$$ of both is an irreducible $$G$$-module. My question is, whether this property is preserved under taking the symmetric power:

Let $$W$$ be an irreducible $$\textrm{O}(n)$$-submodule of $$\textrm{Sym}^p(V)$$. Is it true that $$\textrm{diag}^\vee(W)$$ is irreducible as $$G$$-module?