The generating groups of geometrically uniform spherical signal sets (Q1205128)

An \([M,n]\) spherical signal set is a collection \(\mathcal S\) of \(M\) unit norm vectors in the Euclidean \(n\)-dimensional space \({\mathcal R}^ n\). Its configuration matrix \(C\) is the matrix of the scalar products between pairs of vectors. \(\mathcal S\) is geometrically uniform if, given any two vectors \(x_ i\), \(x_ j\in{\mathcal S}\) there exists an isometry that transforms \(x_ i\) to \(x_ j\) while leaving \(\mathcal S\) invariant. A generating group of \(\mathcal S\) is a group of isometries of \({\mathcal R}^ n\) that transforms any given vector of \(\mathcal S\) into each of the vectors of \(\mathcal S\) while leaving \(\mathcal S\) invariant. The paper characterizes the configuration matrix of a geometrically uniform spherical signal set and gives an algorithm for determining generating groups. The following theorem in particular is established: Let \(C\) be the configuration matrix of an \([M,n]\) spherical signal set. The set is geometrically uniform if and only if (i) The rows of \(C\) are permutations of the first one (ii) The matrix \(J\), all of whose elements are unity and whose order \(j\) is not greater than \((M-1)\)! exists such that \(C=C\otimes J\) commutes with all the matrices of a right regular representation of a group \(\mathcal G\) with \(|{\mathcal G}|=Mj\).