Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras

TL;DR

The paper explicitly constructs complete sets of primitive orthogonal idempotents in symmetric powers of quaternion and octonion algebras, distinguishing cases based on parity of n.

Contribution

It provides explicit formulas and constructions for primitive orthogonal idempotents in symmetric tensor powers of quaternion and octonion algebras, including complexified and associative subalgebras.

Findings

Explicit sets of primitive orthogonal idempotents for symmetric powers of quaternion algebras.

Explicit sets of primitive orthogonal idempotents for symmetric powers of octonion algebras.

Formulas differ for even and odd n, with detailed counts and constructions.

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 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).