Boolean invertible matrices identified from two permutations and their corresponding Haar-type matrices (Q2431172)

Consider a Boolean invertible \(m\times m\) matrix~\(R\), whose rows \(R_1,\dots,R_m\) satisfy \( R_i\circ R_j=c_{ij}R_{\max{(i,j)}},\, c_{ij}\in\{0,1\},\, i,j=1,\dots,m. \) Here \(\circ\) denotes the Hadamard product. The authors prove that \(R\) is uniquely determined by a pair of permutations of~\(\{1,\dots,m\}\). They also identify Haar-type matrices by a pair of permutations. A Haar-type matrix is a real orthogonal \(m\times m\) matrix \(H=(h_{ij})\), whose rows \(H_1,\dots,H_m\) satisfy i)~\(H_i\circ H_j=c_{ij}H_{\max{(i,j)}},\, c_{ij}\in\mathbb{R},\, i,j=1,\dots,m\), \(i\neq j\), ii)~\(H_1=m^{-\frac{1}{2}}(1,\dots,1)\) and each of \(H_2,\dots,H_m\) contains only two entries with alternate signs, iii)~\(\sum_{j=1}^mh_{ij}=0\), \(i=2,\dots,m\).