Gorenstein matrices

• Published in: Algebra and Discrete Mathematics
• Date: 2005
• Main Authors: Dokuchaev, M.A., Kirichenko, V.V., Zelensky, A.V., Zhuravlev, V.N.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2005
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/156609
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Gorenstein matrices / M.A. Dokuchaev, V.V. Kirichenko, A.V. Zelensky, V.N. Zhuravlev // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 8–29. — Бібліогр.: 24 назв. — англ.

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 .