Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras

Abstract

We explicitly find a complete set of $\frac{1}{4}(n+2{)}^2$ (resp. $\frac{1}{4}(n+1)(n+3)$) primitive orthogonal idempotents in ${\mathrm{Sym}}^n\mathbb{H}{\otimes }_{\mathbb{R}}\mathbb{C}$ if $n$ is even (resp. odd), where ${\mathrm{Sym}}^n\mathbb{H}$ is the $n^{th}$ symmetric power of the Hamilton quaternion algebra $\mathbb{H}$. We also give a complete set of $\frac{1}{4}(n+2{)}^2$ (resp. $\frac{1}{8}(n+1)(n+3)$) primitive orthogonal idempotents in ${\mathrm{Sym}}^n\mathbb{H}$ if $n$ is even (resp. odd). Moreover, we explicitly find a complete set of $\frac{1}{24}(n+2)(n+3)(n+4)$ (resp. $\frac{1}{24}(n+1)(n+3)(n+5)$) primitive orthogonal idempotents in the associative subalgebra $({\mathrm{Sym}}^n\mathbb{H}\cdot Z({\mathrm{Sym}}^n\mathbb{O})){\otimes }_{\mathbb{R}}\mathbb{C}$ of ${\mathrm{Sym}}^n\mathbb{O}{\otimes }_{\mathbb{R}}\mathbb{C}$ if $n$ is even (resp. odd), where ${\mathrm{Sym}}^n\mathbb{O}$ is the $n^{th}$ symmetric power of the Cayley octonion algebra $\mathbb{O}$ and $Z({\mathrm{Sym}}^n\mathbb{O})$ is its center. We also give a complete set of $\frac{1}{24}(n+2)(n+3)(n+4)$ (resp. $\frac{1}{48}(n+1)(n+3)(n+5)$) primitive orthogonal idempotents in the associative subalgebra ${\mathrm{Sym}}^n\mathbb{H}\cdot Z({\mathrm{Sym}}^n\mathbb{O})$ of ${\mathrm{Sym}}^n\mathbb{O}$ if $n$ is even (resp. odd).