The isomorphism problem for linear representations and their graphs

TL;DR

This paper investigates the isomorphism problem for linear representations and their associated graphs, establishing conditions under which isomorphisms correspond to collineations and describing automorphism groups explicitly.

Contribution

It provides new criteria linking isomorphisms of linear representations to PGammaL-equivalence of point sets and characterizes automorphism groups induced by collineations.

Findings

Isomorphism of linear representations corresponds to PGammaL-equivalence of point sets.

Automorphism groups are explicitly described in terms of collineations.

Conditions are identified under which automorphisms are induced by ambient space collineations.

Abstract

In this paper, we study the isomorphism problem for linear representations. A linear representation Tn*(K) of a point set K is a point-line geometry, embedded in a projective space PG(n+1,q), where K is contained in a hyperplane. We put constraints on K which ensure that every automorphism of Tn*(K) is induced by a collineation of the ambient projective space. This allows us to show that, under certain conditions, two linear representations Tn*(K) and Tn*(K') are isomorphic if and only if the point sets K and K' are PGammaL-equivalent. We also deal with the slightly more general problem of isomorphic incidence graphs of linear representations. In the last part of this paper, we give an explicit description of the group of automorphisms of Tn*(K) that are induced by collineations of PG(n+1,q).