The Dual Vector Spaces of Symmetric Tensor Powers of Composition Algebras

Let Formula: see text be a quaternion algebra or an octonion algebra over a field Formula: see text of characteristic Formula: see text. In a previous work, we show that the Formula: see textth symmetric power Formula: see text of Formula: see text is isomorphic to a direct sum of central simple algebras Formula: see text, Formula: see text. In this work, we study the dual vector spaces of the Formula: see text’s. We show that Formula: see text, under the isomorphism induced by the isomorphism Formula: see text, where Formula: see text is the Formula: see text-linear map Formula: see text defined by Formula: see text for Formula: see text, with Formula: see text and Formula: see text being the normalized trace and involution, respectively, of Formula: see text.