Hot Lunch
Master Black Belt
I'm not a fan of martial arts rank conferring any sort of authority over others*.
There is a limited sort of "authority" inherent in being the owner of a school or a teacher of a class. The school owner gets to decide who can attend the school. The instructor in a class gets to decide what material is being taught in class that day, how the class time is structured, and can kick out someone who is being disruptive. (Although I've never had to do that last one so far in all my years of teaching.) But that's down to the role of owner or instructor, not rank. If I (BJJ 3rd degree black belt), attend a class taught by a brown belt, then I am there as a student. I do the same drills and exercises that the teacher has everyone else do. I certainly don't try to override their authority as instructor based on my own rank.
*I'm interpreting the discussion so far regarding authority as meaning "having authority over others" rather than "being an authority (expert) in a subject."
I was talking about this earlier. In ISKF (Shotokan), instructor, judge, and examiner qualifications are separate from rank. There are minimum ranks required for each level of these qualifications, but they are obtained separately from the rank. You can be a nidan and have some of these qualifications, while you can be a yondan and have none.
But, in this case, authority would come with the position and not the rank. Either way, it's not conditional on how well received you are by those under you.