We study the subgroup structure of the semigroup of real square matrices of given dimension under tropical matrix multiplication. We show that every maximal subgroup is isomorphic to the full linear automorphism group of a related tropical polytope, and that each of these groups is the direct product of R with a ﬁnite group. We also show that there is a natural and canonical embedding of each full rank maximal subgroup into the group of units of the semigroup. Out results have numerous corollaries, including the fact that every automorphism of a full rank projective tropical polytope extends to an automorphism of the containing space, and that every full rank subgroup has a common eigenvector.
Bibliographical noteZur Izhakian: Research supported by the Alexander von Humboldt Foundation. Marianne Johnson: Research supported by EPSRC Grant EP/H000801/1. Mark Kambites: Research supported by EPSRC Grant EP/H000801/1. Mark Kambites gratefully acknowledges the hospitality of Universität Bremen during a visit to Bremen.
- Tropical matrices
- Green's relations
- tropical polytopes
- automorphism group