On the dimension of Kirichenko space

Abstract

We introduce a notion of the Kirichenko space which is connected with the notion of Gorenstein matrix (see [2], ch. 14). Every element of Kirichenko space is an \(n\times n\) matrix, whose elements are solutions of the equations \(a_{i,j}+a_{j,\sigma (i)} =a_{i,\sigma (i)}\);  \(a_{1,i}=0\) for \(i,j =1,\ldots, n\) determined by a permutation \(\sigma\) which has no cycles of the length \(1\). We give a formula for the dimension of this space in terms of the cyclic type of \(\sigma\).