Lattices and Norton algebras of Johnson, Grassmann and Hamming graphs

Abstract

To each of the Johnson, Grassmann and Hamming graphs we associate a lattice and characterize the eigenspaces of the adjacency operator in terms of this lattice . We also show that each level of the lattice induces in a natural way a tight frame for each eigenspace. For the most important eigenspace we compute explicitly the constant associated to the tight frame. Using the lattice we also give a formula for the product of the Norton algebra attached to that eigenspace.