Construction of vector lists and isomorph rejection

Abstract

It is observed that if Δ is a system of orbit representatives for the action of a finite group G on a G-stable subset L of a finite set S and if F is a field of characteristic zero, then F S is an algebra of dimension | S|. Furthermore, if S = R D , the set of functions from a finite set D to a finite set R, then F S has a multilinear structure. A general problem is stated: Given linear operators T 1 and T 2, construct vectors v 1, … v t ϵ F S such that T 2 I Δ = T 1( v 1 + … + v t ) where l Δ is the indicator or characteristic function of Δ. (Note that this construction gives, in some sense, a solution to the problem of isomorph rejection.) For appropriate choices of T 1 and T 2, two approaches to this construction problem are considered. These are the principle of inclusion-exclusion and backtrack computer programming. In particular, these approaches are discussed when S = R D and the vectors to be constructed are pure or homogeneous tensors.