When we consider Achilles as a tragic hero, we must think about the extent to which his own personality flaws lead to his downfall. Greek mythology certainly explores the role of fate in human life, and we often see gods interfering with events in the human world, using people as pawns in their own games and battles. However, Madeline Miller's The Song of Achilles humanizes these well-known mythical figures, and in doing so, Miller implies that the showdown between Achilles and Agamemnon is the result of two powerful men's personality flaws.
The conflict is rooted in Briseis, a woman who Achilles claims as a war prize and Agamemnon later takes from him in retaliation for Achilles's refusal to obey Agamemnon as his commander. As a result, Achilles pledges not to report to battle and stays in his tent while the Greeks suffer massive losses. His lover and companion, Patroclus, dons Achilles's armor as a decoy but ends up being killed by Hector. Then Achilles famously battles Hector, defeating him and dragging his body around the battleground in a sign of the ultimate disrespect. Achilles's actions in the Trojan War depict him as strong but arrogant, an ultimately flawed man.
Miller's novel implies that Achilles's conflict with Agamemnon stems from this personality flaw. Achilles has become so powerful and renowned by this point in the novel (and this point in the Trojan War) that he is proud and arrogant. Agamemnon, the leader of the army, is similarly strong-willed and wants to assert his dominance over Achilles. Though Achilles is the prize warrior for the Greeks, Agamemnon is the king whose runaway wife, Helen, is at stake in this conflict. Agamemnon's pride has been seriously bruised by his wife's affair with Paris, so he feels it is essential to maintain control over his army and that means maintaining control over his most important asset, Achilles. Each man's desire to be the "alpha" of the Greek troops leads to the conflict, which begins a domino effect that ends with Achilles's own death at the hands of Paris.