Gorenstein matrices

M. A. Dokuchaev, V. V. Kirichenko, A. V. Zelensky, V. N. Zhuravlev

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,n1) 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,,2m1) and

σ(Em)=(1232m12m2m2m12m221).


Keywords


exponent matrix; Gorenstein tiled order, Gorenstein matrix, admissible quiver, doubly stochastic matrix

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