What does the scene in the throne room tell you about each of the three friends: Annabeth, Grover, and Percy?

It shows that Percy is very loyal to his friends and would never leave them behind.
"We Find Out the Truth, Sort of," when they are in Hades thrown room then this scene shows that the friends are willing to sacrifice for each other for the happiness and safety of the others. This also shows the trust that the friends have for one another. Percy had been accused of taking the master bolt, and when it was found in his backpack, both Annabeth and Grover knew and trusted Percy enough to know that he had not taken it. Towards the end of the chapter Percy leaves his mom behind in the underworld with Hades. Both Annabeth and Grover try to persuade Percy to let them stay in the underworld so Percy can have his mom.