Tropical Algebraic Groups ?!

From CRCG-Wiki

Tropical algebraic groups?! (Jan Draisma)

I will report on some work-in-progress, joint with Tyrrell McAllister, on attempts to tropicalise algebraic groups meaningfully. One promising approach is based on Lusztig's totally positive monoid in algebraic groups. I shall explain how one can insert elements of the tropical semifield in the standard generators, and obtain an "abstract totally positive tropical monoid". Canonical bases give rise to representations of this monoid into matrix monoids with tropical multiplication.