Gorenstein matrices

Abstract

Let A = (aij ) be an integral matrix. We say that A is (0, 1, 2)-matrix if aij ∈ {0, 1, 2}. There exists the Gorenstein (0, 1, 2)-matrix for any permutation σ on the set {1, . . . , n} without fixed elements. For every positive integer n there exists the Gorenstein cyclic (0, 1, 2)-matrix An such that inx An = 2. If a Latin square Ln with a first row and first column (0, 1, . . . n − 1) is an exponent matrix, then n = 2m and Ln is the Cayley table of a direct product of m copies of the cyclic group of order 2. Conversely, the Cayley table Em of the elementary abelian group Gm = (2)×. . .×(2) of order 2 m is a Latin square and a Gorenstein symmetric matrix with first row (0, 1, . . . , 2 m − 1) and σ(Em) = 1 2 3 . . . 2 m − 1 2m 2 m 2 m − 1 2m − 2 . . . 2 1 .