Some Remarks on the Non-Abelian Fourier Transform in Crossover Designs in Clinical Trials

Abstract

Let G be a non-abelian group and let l 2 (G) be a finite dimensional Hilbert space of all complex valued functions for which the elements of G form the (standard) orthonormal basis. In our paper we prove results concerning G -decorrelated decompositions of functions in l 2 (G) . These G -decorrelated decompositions are obtained using the G -convolution either by the irreducible characters of the group G or by an orthogonal projection onto the matrix entries of the irreducible representations of the group G . Applications of these G -decorrelated decompositions are given to crossover designs in clinical trials, in particular the William’s 6×3 design with 3 treatments. In our example, the underlying group is the symmetric group S 3 .