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Abstract: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 the associative subalgebra $\big({\rm Sym}^n\mathbb{H}\cdot Z({\rm Sym}^n\mathbb{O})\big)\otimes_{\mathbb{R}}\mathbb{C}$ 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 $\mathbb{O}$ and $Z({\rm Sym}^n\mathbb{O})$ is its center. 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 the associative subalgebra ${\rm Sym}^n\mathbb{H}\cdot Z({\rm Sym}^n\mathbb{O})$ of ${\rm Sym}^n\mathbb{O}$ if $n$ is even (resp. odd).

Submission history

From: Aharon Razon Dr. [view email]
[v1] Sat, 11 Apr 2026 18:54:18 UTC (21 KB)
[v2] Sat, 25 Apr 2026 07:18:20 UTC (21 KB)
[v3] Sat, 13 Jun 2026 20:07:32 UTC (21 KB)
[v4] Thu, 25 Jun 2026 04:44:15 UTC (62 KB)