in authority: holding the power to decide, whether over a kingdom, in an office, or in a single action that is allowed.
This Aramaic word answers one question: who has the say. It can be said of a person, so that the man himself is named as the one with power, and it can be turned into a flat statement about who is in charge, which is what the book of Daniel does with it again and again.
Its widest stretch is the legal one. When the Persian king writes that no one may tax the temple staff, this is the word he uses, and it means that nobody has the standing to do it. Being allowed and being in charge are a single idea here, because permission comes down from whoever holds the power.
The word describes a state rather than reporting an action. The matching Aramaic verb (שְׁלֵט) says that someone ruled or gained power; the matching noun (שָׁלְטָן) names the ruling power itself as a thing that can be handed over, taken away, or made to last forever. This word sits between them and says of a person that the power is his.
In Daniel it carries the book's central claim. Four times the same sentence lands on a proud king: the Most High is the one in charge of human kingdoms, and he gives them to whomever he wants. A fifth line says it in three words, that heaven rules, and it is the sentence Nebuchadnezzar has to learn before he gets his throne back.