Which shared value of the Achaians is violated in Book 1 of Homer's Iliad?
In Book 1 of Homer's Iliad, the quarrel between Agamemnon and Achilles revolves around the concept of honor. Both Agamemnon and Achilles believe that, as active participants in the war, they are entitled to a share of the spoils. This share of the spoils is an outward and visible sign of the honor that the army has bestowed upon a warrior.
In Iliad 1, Agamemnon must give back Chryseis to her father. This causes him to lose the tangible and visible symbol of his honor. Agamemnon feels that, as leader of the Greeks, he should not be without a share of honor. Accordingly, because Achilles emerges as the prime mover behind Agamemnon's return of Chryseis, Agamemnon decides to take Briseis, the symbol of Achilles' honor, away from him.
Thus, once this imbalance in the share of honor is created, Achilles feels that the only way to correct it is to withdraw from the fighting. Achilles decides that it would be better for him "to lead my beaked ships home than stay here dishonoured piling up wealth and goods for you" (A.S. Kline translation). Achilles declares that soon Agamemnon will be sorry "for failing to honour the best of the Achaeans."
From a modern perspective, both Agamemnon and Achilles seem to behave badly: Agamemnon for acting so arrogantly and Achilles for seeming to act like a baby just because his war prize has been taken away.
From the Homeric perspective, though, I imagine that Agamemnon is regarded as being most in the wrong, since Agamemnon has dishonored Achilles in front of the entire army.
Fortunately for Achilles, though, he has friends in high places. His mother gets Zeus to agree to restore Achilles' honor:
For Agamemnon the king shows disrespect, arrogantly seizing his rightful prize. Avenge my son, Olympian Zeus, lord of justice; enhance the Trojans’ power, till the Greeks honour and respect my son and make amends. (A.S. Kline translation)
Agamemnon wronged Achilles by despoiling him of his fair share of the pillage: Briseis. this should in no way be done by any soldier to any other, let alone to a demi-god like Achilles. It's inacceptable especially by the Greeks a warlike people at the time.
Achilles offended the king by his vindictive. Being the king of men, Agamemnon should in no way be spoken to so roughly and harshly as Achilles did.