Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras

We explicitly find a complete set of 14(n+2)² (resp. 14(n+1)(n+3)) primitive orthogonal idempotents in SymⁿH⊗RC if n is even (resp. odd), where SymⁿH is the nᵗʰ symmetric power of the Hamilton quaternion algebra H. We also give a complete set of 14(n+2)² (resp. 18(n+1)(n+3)) primitive orthogonal idempotents in SymⁿH if n is even (resp. odd). Moreover, we explicitly find a complete set of 124(n+2)(n+3)(n+4) (resp. 124(n+1)(n+3)(n+5)) primitive orthogonal idempotents in the associative subalgebra ( SymⁿH· Z( SymⁿO))⊗RC of SymⁿO⊗RC if n is even (resp. odd), where SymⁿO is the nᵗʰ symmetric power of the Cayley octonion algebra O and Z( SymⁿO) is its center. We also give a complete set of 124(n+2)(n+3)(n+4) (resp. 148(n+1)(n+3)(n+5)) primitive orthogonal idempotents in the associative subalgebra SymⁿH· Z( SymⁿO) of SymⁿO if n is even (resp. odd).