The structure of symmetric tensor powers of composition algebras

Let C be a composition algebra which is either the Hamilton quaternion algebra H or the Cayley octonion algebra O over R. In a previous work, the nth symmetric power Sym(n)C of C is shown to be a direct sum of central simple algebras, corresponding to the partitions of n of length 2, such that the component corresponding to the partition (m, n - m) is isomorphic to the component T2m-n C of Sym(2m-n)C corresponding to the partition (2m - n, 0) of 2m - n. In this work, we study the building blocks TnC of these decompositions. We show that the "local" structure of C, i.e. the complex-like subfields of C, determine both the complement of TnC in Sym(n)C and the trace map of TnC, induced from the trace map of C. We also derive a recursive trace formula on the TnC's. We use the "local-global" results to define positive definite symmetric bilinear forms on the vector space circle plus(infinity)(n=0) TnC, which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra TnC is described.