Symmetrical $2$-extensions of the $2$-dimensional grid. II

E. A. Konoval'chik ; K. V. Kostousov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 192-211

Résumé

The investigation of symmetrical $q$-extensions of a $d$-dimensional cubic grid $\Lambda^{d}$ is of interest both for group theory and for graph theory. For small $d\geq 1$ and $q>1$ (especially for $q=2$), symmetrical $q$-extensions of $\Lambda^{d}$ are of interest for molecular crystallography and some phisycal theories. Earlier V. Trofimov proved that there are only finitely many symmetrical $2$-extensions of $\Lambda^{d}$ for any positive integer $d$. This paper is the second and concluding part of our work devoted to the description of all, up to equivalence, realizations of symmetrical $2$-extensions of $\Lambda^{2}$ (we show that there are $162$ such realizations). In the first part of our work, which was published earlier, we found all, up to equivalence, realizations of symmetrical $2$-extensions of $\Lambda^{2}$ such that only the trivial automorphism fixes all blocks of the imprimitivity system ($87$ realizations). In the present paper, we find the remaining realizations of symmetrical $2$-extensions of $\Lambda^{2}$.

Détail

Keywords: symmetrical extension of a graph, $d$-dimensional grid.