Finite tight frames for the eigenspaces of some Laplacian operators

FERNANDO LEVSTEIN; CAROLINA MALDONADO; DANIEL PENAZZI

Córdoba

Workshop; VII workshop in Lie theory and its applications; 2009

FaMAF UNC

Let F_q be a finite field with q elements and B a non-degenerate bilinear form on F_q^n. A subspace W is isotropic if B(x,y)=0 for all x,y in W. We consider the group G C GL(n, F) of linear transformations leaving the form B invariant. G acts transitively on the set X of maximal isotropic subspaces of F^n. The representation of G on L(X)={f:X --> C} is multiplicity free and we have the decomposition of L(X) as a sum of irreducible representations V_i. A tight frame {v_j}_{j in J} of a subspace U C L(X) is a set of generators of U that satisfy the equation $$v=c\sum_{j\in J} v_j \quad \forall v\in U$$ for some fixed scalar c and < , > the standard hermitian form on L(X). The set X has a distance given by d(x, y) = dim(x)- dim(x\cap y) and each V_i is an eigenspace of the Laplacian operator on L(X): $$\Delta f(x)= \sum_{y: d(x,y)=1} f(y)$$ The problem we considered was to give a nice basis to each of the V_i. Instead of a basis we were able to find a corresponding tight frame T_i. As an application we give an explicit formula for the Norton algebra product on the V_i corresponding to the second largest eigenvalue of the Laplacian.