I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear about your situation.
If you have reason to believe that you're getting passed over for a promotion because of your race, ethnicity, religion, gender, sexual orientation, or age, that would be illegal. However, if the reason you're not getting promoted is based on a personality conflict, I'm sorry to say, that IS perfectly legal. There isn't any anti-discrimination law that protects you in that case, because there's no protected class being discriminated against. There's no legal right to a promotion, so there isn't any cause of action when you don't get one.
The only recourse here is to go up the chain of command, talk to HR, and see if there's anything that you can do. Maybe reach out to the Director to see if there's anything you can do. But I'm afraid that legally, there's no way to make them give you the promotion based on these facts.
I apologize that this was probably not the Answer you were hoping to receive. However, it would be unfair to you and unprofessional of me were I to provide you with anything less than truthful and honest information. I hope you understand.