  This paper deals with the problem of factorizing integer powers of the
Laplace operator acting on functions taking values in higher spin
representations. This is a far-reaching generalization of the well-known fact
that the square of the Dirac operator is equal to the Laplace operator. Using
algebraic properties of projections of Stein-Weiss gradients, i.e. generalized
Rarita-Schwinger and twistor operators, we give a sharp upper bound on the
order of polyharmonicity for functions with values in a given representation
with half-integral highest weight.
