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The isomorphism problem for linear representations and their graphs

Journal Contribution - Journal Article

In this paper, we study the isomorphism problem for {\em linear representations}. A linear representation $T_n^*(\K)$ 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 $T_n^*(\K)$ is induced by a collineation of the ambient projective space. This allows us to show that, under certain conditions, two linear representations $T_n^*(\K)$ and $T_n^*(\K')$ 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 {\em incidence graphs} of linear representations.

In the last part of this paper, we give an explicit description of the group of automorphisms of $T_n^*(\K)$ that are induced by collineations of $\PG(n+1,q)$.
Journal: Advances in Geometry
ISSN: 1615-715X
Issue: 14
Pages: 353-367
Publication year:2013
Keywords:linear representation, incidence graph, automorphism group
  • ORCID: /0000-0001-8230-4451/work/62230785