Explicit complete sets of primitive orthogonal idempotents are constructed in Sym^n H tensor C, Sym^n H, and certain subalgebras of Sym^n O tensor C and Sym^n O, with counts depending on parity of n.
Razon,The structure of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol
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Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras
Explicit complete sets of primitive orthogonal idempotents are constructed in Sym^n H tensor C, Sym^n H, and certain subalgebras of Sym^n O tensor C and Sym^n O, with counts depending on parity of n.