Tropical matrix groups

Zur Izhakian, Marianne Johnson, Mark Kambites

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Abstract

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.
Original languageEnglish
Pages (from-to)178-196
Number of pages19
JournalSemigroup Forum
Volume96
Issue number1
Early online date14 Sep 2017
DOIs
Publication statusPublished - Feb 2018

Fingerprint

Matrix Groups
Maximal Subgroup
Polytope
Automorphism
Semigroup
Subgroup
Group of Units
Matrix multiplication
Linear Group
Square matrix
Direct Product
Automorphism Group
Eigenvector
Corollary
Isomorphic

Keywords

  • Tropical matrices
  • semigroups
  • Green's relations
  • tropical polytopes
  • automorphism group

Cite this

Tropical matrix groups. / Izhakian, Zur; Johnson, Marianne; Kambites, Mark.

In: Semigroup Forum, Vol. 96, No. 1, 02.2018, p. 178-196.

Research output: Contribution to journalArticle

Izhakian, Z, Johnson, M & Kambites, M 2018, 'Tropical matrix groups', Semigroup Forum, vol. 96, no. 1, pp. 178-196. https://doi.org/10.1007/s00233-017-9894-1
Izhakian, Zur ; Johnson, Marianne ; Kambites, Mark. / Tropical matrix groups. In: Semigroup Forum. 2018 ; Vol. 96, No. 1. pp. 178-196.
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