### Gorenstein matrices

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

#### Abstract

Let $$A=(a_{ij})$$ be an integral matrix. We say that  $$A$$ is $$(0, 1, 2)$$-matrix if $$a_{ij}\in \{0, 1, 2\}$$. There exists the Gorenstein $$(0, 1, 2)$$-matrix for any permutation $$\sigma$$ on the set $$\{1, \ldots , n\}$$ without fixed elements. For every positive integer $$n$$ there exists the Gorenstein cyclic $$(0, 1, 2)$$-matrix $$A_{n}$$ such that $$inx\,A_{n}=2$$.

If a  Latin square $${\mathcal L}_{n}$$ with a first row and first column $$(0, 1,\ldots n-1)$$ is an exponent matrix, then $$n=2^{m}$$ and $${\mathcal L}_{n}$$ is the Cayley table of a direct product of $$m$$ copies of the cyclic group of order 2. Conversely, the Cayley table $${{\mathcal E}}_{m}$$ of the elementary abelian group $$G_{m}=(2)\times\ldots \times (2)$$  of  order $$2^{m}$$ is a Latin square and a Gorenstein symmetric matrix with  first row $$(0, 1,\ldots , 2^{m}-1)$$ and

$$\sigma({{\mathcal E}}_{m})=\begin{pmatrix} 1&2&3&\ldots &2^{m}-1&2^{m}\\ 2^{m}&2^{m}-1&2^{m}-2&\ldots & 2&1\end{pmatrix}.$$

#### Keywords

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

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