On the algebras of symmetries (groups of collineations) of designs from finite Desarguesian planes with applications in statistics

Abstract

Decomposition into a direct sum of irreducible representations of the representation of the full collineation group of a finite Desarguesian plane, as a group of matrices permuting the flags of the plane and the simple components of the corresponding commutant algebra, have been worked out here for the projective plane PG(2, 2) and the affine plane EG(2, 3). The dimension and the components of the covariance matrix of the observations from a design derived from such a plane, which commutes with such a permutation representation of the full collineation group of the plane, are thus determined. This paper is in the spirit of earlier works by, James (1957), Mann (1960), Hannan (1961, 1965), McLaren (1963), and Sysoev and Shaikan (1976). A. T. James, Ann. Math. Statist. 28 (1957), 993–1002, H. B. Mann, Ann. Math. Statist. 31 (1960), 1–15, E. J. Hannan, Research Report (Part. (I)), Summer Research Institute, Australian Math. Soc. and Methuen's Monographs on Applied Probability and Statistics, Supplementary Review Series in Applied Probability, Vol. 3, A. D. McLaren, Proc. Cambridge Philos. Soc. 59 (1963), 431–450, and L. P. Sysoev and M. E. Shaikin, Avtomat. i Telemekh. 5 (1976), 64–73.