Lattices, frames and Norton algebras of dual polar graphs

Abstract

To a dual polar graph (X; E) we associate a graded lattice, map the lattice onto L2(X) and characterize the eigenspaces of the adjacency operator Δ in L2(X) in terms of this map, each one corresponding to the levels of the lattice. The map also induces in a natural way a tight frame on each eigenspace of Δ, and we find the constants associated to each tight frame. As an application we give a formula for the product of the Norton algebra of the eigenspace associated to the second largest eigenvalue of Δ.