The dual vector spaces of symmetric tensor powers of composition algebras

Let C be a quaternion algebra or an octonion algebra over a field F of characteristic 0. In a previous work, we show that the nth symmetric power SymnC of C is isomorphic to a direct sum of central simple algebras (SmC)-C-(n), m = (sic)n/2 (sic), . . . , n. In this work, we study the dual vector spaces of the (SmC)-C-(n)'s. We show that ((SmC)-C-(n))* congruent to Sym(2m-n)(C*)mu(n-m)/Sym(2m-n-2)(C*)mu(n-m+1), under the isomorphism induced by the isomorphism (C-circle times n)* congruent to (C*)(circle times n), where mu is an element of (C*)(circle times 2) = (C-circle times 2)* is the F-linear map C circle times C -> F defined by mu(a circle times b) = Ntr((a) over barb) for a,b is an element of C, with Ntr and (-) being the normalized trace and involution, respectively, of C.