ec 2 00 3 SYMMETRIC FUNCTIONS AND THE PHASE PROBLEM IN CRYSTALLOGRAPHY

The calculation of crystal structure from X-ray diffraction data requires that the phases of the " structure factors " (Fourier coefficients) determined by scattering be deduced from the absolute values of those structure factors. Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case of crystal structures with n equal atoms in the unit cell, with n small. We rephrase the problem as a question about multiplicative in-variants for a particular finite group action. We show that the absolute values form a generating set for the field of invariants of this action, and consider the problem of making this theorem constructive and practical; the most promising approach for deriving explicit formulas uses SAGBI bases.