- Why doesn't Achilles simply kill Agamemnon and his army?
Agamemnon seriously disses him, repeatedly and it's an honor/shame society.
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Because Athena forbids it, and says he will be rewarded if he obeys. From the Iliad book 1:
While Agamemnon spoke, the son of Peleus was gnawed by pain, and the heart in his shaggy breast was torn; whether to draw the sharp blade at his side, scatter the crowd, and kill the son of Atreus, or curb his wrath and restrain his spirit. As he pondered this in his mind, his great sword half-unsheathed, Athene descended from the sky, sent by Hera, the white-armed goddess, who loved and cared for both the lords alike. Athene, standing behind the son of Peleus, tugged at his golden hair, so that only he could see her, no one else. Achilles, turning in surprise, knew Pallas Athene at once, so terrible were her flashing eyes. He spoke out, with winged words, saying: ‘Why are you here, daughter of aegis-bearing Zeus? Is it to witness Agamemnon’s arrogance? I tell you and believe that this son of Atreus’ will pay soon with his life for his insolent acts.’
The goddess, bright-eyed Athene, replied: ‘I came from the heavens to quell your anger, if you’ll but listen: I was sent by the goddess, white-armed Hera, who in her heart loves and cares for you both alike. Come, end this quarrel, and sheathe your sword. Taunt him with words of prophecy; for I say, and it shall come to pass, that three times as many glorious gifts shall be yours one day for this insult. Restrain yourself, now, and obey.’
Then swift-footed Achilles, in answer, said: ‘Goddess, a man must attend to your word, no matter how great his heart’s anger: that is right. Whoever obeys the gods will gain their hearing.’
The whole honor/shame dichotomy is a big question, but it's not quite accurate to say that the Greeks simply killed whomever was shaming them. That's not how it works. The point of honor/shame is that Achilles needs to do what will bring him honor, and there is no honor in killing the king - attacking above your station is actually a form of hubris and would only bring him more shame. Shame is not internal: it's what others think of him. Disregarding the heroic code is what is shameful, as is seen from Ajax's words in the embassy scene.
With this he signaled to Patroclus, with a nod of his head, to spread a comfortable bed for Phoenix, so the others might take the hint and leave the hut. Ajax, godlike son of Telamon, then spoke: ‘Odysseus of many wiles, Zeus-born son of Laertes, we should go, since we achieve nothing by staying here. We must hasten to give the news to the Greeks, who no doubt await us, bad though it is. Achilles’ proud heart has raised him to such a pitch of fury he forgets, harsh man that he is, his comrades’ love, with which we in the fleet honoured him above all others. He shows no pity! Yet a man accepts blood-money even from his brother’s or his son’s killer, and the killer is not expelled from the land if he pays the price to the next of kin, whose pride and feelings are appeased by such compensation. Achilles, the gods have hardened and poisoned your heart, all because of a girl, while we offer you seven, the best there are by far, and a host of gifts besides. So be gracious and show respect for your house, since we represent the Danaans here, and are keen to remain your closest and dearest friends of all the Achaeans.’
His refusal to concede is actually what is bringing him dishonor, and if he had made amends and gone back into battle, he would have been honored even more.