Polynomial spaces (Q1812526)

This important paper generalizes techniques from association schemes and spherical designs to study finite subsets of polynomials spaces. A polynomial space is a set \(\Omega\) with a function \(\rho: \Omega\times\Omega\to {\mathbf R}\) with \(\rho(x,x)=r>0\), \(\rho(x,y)=\rho(y,x)<r\) for \(x\neq y\), such that the zonal functions \(x\to \rho(a,x)\) \((a\in \Omega)\) span a finite-dimensional space. This defines the minimal requirements needed to discuss \(t\)-designs, bounds on subsets with few values of \(\rho\), and relations to association schemes. Particularly interesting is the fact that \(t\)-transitive permutation groups can be viewed (via Burnside's lemma) as \(t\)-designs in the symmetric group.