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 text]th 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].