An incidence algebra for t-designs with automorphisms
A t-design or generalized Steiner system S(\(\lambda\) ;t,k,v) is a pair (X,\({\mathcal B})\) with a v-set X of points and a family \({\mathcal B}\) of k- subsets of X called blocks such that, each block has k points and any t points are contained in exactly \(\lambda\) blocks. An automorphism of (X,\({\mathcal B})\) is a permutation of X which preserves \({\mathcal B}\). In this paper the algebra of matrices, whose rows and columns are indexed by the members of \({\mathcal P}(X)\), that are invariant under the natural action of a group \(G\leq Sym(X)\) is introduced. An epimorphism \(\tau\) from this algebra onto the matrices whose rows and columns are indexed by the orbits of G acting on \({\mathcal P}(X)\) is discovered. This mapping carries the matrices of Wilson onto the matrices of Kramer and Mesner and therefore can be used to generalize the t-design inequalities of Fisher, Wilson, and Ray-Chaudhuri. A conjecture of Earl Kramer is settled and an elementary proof of a theorem of Livingston and Wagner is presented.
scientific article; zbMATH DE number 4057522
scientific article