ON THE STABILIZATION OF THE NUMBER OF ORBITS OF HIGH-RANK CAMERON MATRICES

Abstract

A quadratic (0,1)-matrix of degree n with just n units among its elements will be called a Cameron matrix. The orbits of the natural action of the group S_n× S_n (the square of the symmetric group of degree n) on the set of Cameron matrices of degree n (an independent action on the rows and the columns of matrices) are considered. It is proved that for fixed d < n, the number of such orbits for matrices of rank n – d is constant for n ≥ 3d and grows with the growth of n if n < 3d. For each orbit, its representative in a quasi-Jordan form is indicated.