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 inxAn=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 2m is a Latin square and a Gorenstein symmetric matrix with first row (0,1,…,2m−1) and
σ(Em)=(123…2m−12m2m2m−12m−2…21).
Keywords
exponent matrix; Gorenstein tiled order, Gorenstein matrix, admissible quiver, doubly stochastic matrix
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