Supertropical $\operatorname{SL}_n$

TL;DR

This paper develops a symmetric supertropical version of the special linear group, partitioning it into submonoids, analyzing its structure, and exploring the action of elementary matrices, revealing novel behaviors in supertropical algebra.

Contribution

It introduces a supertropical $ ext{SL}_n$ version, partitions it into submonoids, and studies its structure and matrix actions, extending supertropical algebra theory.

Findings

Partition into submonoids based on quasi-identity matrices

Identification of maximal sub-semigroups of $ ext{SLS}_n$

Illustration of unexpected behaviors in supertropical matrix actions

Abstract

Extending earlier work on supertropical adjoints and applying symmetrization, we provide a symmetric supertropical version $\operatorname {SLS}_n$ of the special linear group, which we partition into submonoids, based on "quasi-identity" matrices, and we display maximal sub-semigroups of $\operatorname {SLS}_n$. We also study the monoid generated by $\operatorname {SLS}_n$. Several illustrative examples are given of unexpected behavior. We describe the action of elementary matrices on $\operatorname {SLS}_n$, which enables one to connect different matrices in $\operatorname {SLS}_n$, but in a weaker sense than the classical situation.