The structure of symmetric tensor powers of composition algebras

Abstract

Let [Formula: see text] be a composition algebra which is either the Hamilton quaternion algebra [Formula: see text] or the Cayley octonion algebra [Formula: see text] over [Formula: see text]. In a previous work, the nth symmetric power [Formula: see text] of [Formula: see text] is shown to be a direct sum of central simple algebras, corresponding to the partitions of [Formula: see text] of length [Formula: see text], such that the component corresponding to the partition [Formula: see text] is isomorphic to the component [Formula: see text] of [Formula: see text] corresponding to the partition [Formula: see text] of [Formula: see text]. In this work, we study the building blocks [Formula: see text] of these decompositions. We show that the “local” structure of [Formula: see text], i.e. the complex-like subfields of [Formula: see text], determine both the complement of [Formula: see text] in [Formula: see text] and the trace map of [Formula: see text], induced from the trace map of [Formula: see text]. We also derive a recursive trace formula on the [Formula: see text]’s. We use the “local-global” results to define positive definite symmetric bilinear forms on the vector space [Formula: see text], which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra [Formula: see text] is described.