On third order rotatability

Third order rotatability of experimental designs, moment matrices and information surfaces is investigated, using a Kronecker power representation. This representation complicates the model but greatly simplifies the theoretical development, and throws light on difficulties experienced in some previous work. Third order rotatability is shown to be characterized by the finitely many transformations consisting of permutations and a bi-axial 45 degree rotation, and the space of rotatable third order symmetric matrices is shown to be of dimension 20, independent of the number of factorsm. A general Moore-Penrose inverse of a third order rotatable moment matrix is provided, leading to the information surface, and the corresponding optimality results are discussed. After a brief literature review, extensions to higher order models, the connections with tensor representations of classic matrix groups, and the evaluation of a general dimension formula, are all explored.