Two thumbs up! No, down! I meant down!!

How did Roman emperors order the death of a gladiator?

Thumbs up.

Neither Roman spectators calling for the death of a gladiator, nor Roman emperors authorizing one ever gave a thumbs down. In fact, the Romans didn't use a thumbs down sign at all.

If death was desired, the thumb was stuck up-like a drawn sword. For a loser's life to be spared, the thumb was tucked away inside the closed fist-as with a sheathed weapon. This is expressed in Latin as "pollice compresso favor iudicabatur" (goodwill is decided by the thumb being kept in).

So yes, the movie The Gladiator is incorrect when it shows Commodus giving the thumbs up to spare Maximus. Ridley Scott was aware of the misconception, but did not want to confuse the audience.