The relationship between Achilles and Agamemnon is that Achilles is an independent prince who owes him nothing.
Both of these two personalities are from very different backgrounds. The relationship that they both had was the war that they fought in Troy.
They both fell out over a woman named Briseis. Achilles was fond of her but she was taken away from him by King Agamemnon. These led to issues that made him withdraw his troops from the war.
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