Factorization of Laplace operators on higher spin representations

TL;DR

This paper generalizes the factorization of Laplace operators to functions valued in higher spin representations, providing algebraic methods and bounds for polyharmonicity related to these advanced operators.

Contribution

It introduces a novel algebraic approach to factorize powers of the Laplace operator on higher spin valued functions, extending classical results to more complex representations.

Findings

Established sharp upper bounds on polyharmonicity order for higher spin functions

Developed algebraic techniques based on Stein-Weiss gradients and twistor operators

Generalized classical Laplace operator factorization to higher spin contexts

Abstract

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.