Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras

We explicitly find a complete set of ${1\over4}(n+2)^2$ (resp. ${1\over4}(n+1)(n+3)$) primitive orthogonal idempotents in ${\rm Sym}^n\mathbb{H}\otimes_\mathbb{R}\mathbb{C}$ if $n$ is even (resp. odd), where ${\rm Sym}^n\mathbb{H}$ is the $n^{\it th}$ symmetric power of the Hamilton quaternion algebra $\mathbb{H}$. We also give a complete set of ${1\over4}(n+2)^2$ (resp. ${1\over8}(n+1)(n+3)$) primitive orthogonal idempotents in ${\rm Sym}^n\mathbb{H}$ if $n$ is even (resp. odd). Moreover, we explicitly find a complete set of ${1\over24}(n+2)(n+3)(n+4)$ (resp. ${1\over24}(n+1)(n+3)(n+5)$) primitive orthogonal idempotents in a certain associative subalgebra of ${\rm Sym}^n\mathbb{O}\otimes_\mathbb{R}\mathbb{C}$ if $n$ is even (resp. odd), where ${\rm Sym}^n\mathbb{O}$ is the $n^{\it th}$ symmetric power of the Cayley octonion algebra $\mathbb{O}$. We also give a complete set of ${1\over24}(n+2)(n+3)(n+4)$ (resp. ${1\over48}(n+1)(n+3)(n+5)$) primitive orthogonal idempotents in a certain associative subalgebra of ${\rm Sym}^n\mathbb{O}$ if $n$ is even (resp. odd).