What question did this study set out to answer?

The aim is to find complete sets of primitive orthogonal idempotents in symmetric tensor powers of specific composition algebras.

June 28, 2026Communications in Mathematics0 citations

Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras

ARAharon RazonBen-Gurion University of the Negev

Key Points

The aim is to find complete sets of primitive orthogonal idempotents in symmetric tensor powers of specific composition algebras.
Deriving a complete set of primitive orthogonal idempotents using algebraic structures.
Classifying idempotents based on the parity of n, either even or odd.
Exploration of associative subalgebras in relation to symmetric powers.
Identified ${1 ext{/}4}(n+2)^2$ idempotents in ${ m Sym}^n\mathbb{H}\otimes_\mathbb{R}\mathbb{C}$ for even n and ${1 ext{/}4}(n+1)(n+3)$ for odd n.
Found ${1 ext{/}24}(n+2)(n+3)(n+4)$ idempotents in the associative subalgebra ${\rm Sym}^n\mathbb{H}\cdot Z({\rm Sym}^n\mathbb{O})$ for even n; ${1 ext{/}48}(n+1)(n+3)(n+5)$ for odd n.

Abstract

We explicitly find a complete set of 14 (n+2) ² (resp. 14 (n+1) (n+3) ) primitive orthogonal idempotents in SymⁿHRC if n is even (resp. odd), where SymⁿH is the n^ th 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) ) ₑC of SymⁿORC if n is even (resp. odd), where SymⁿO is the n^ th 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).

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