From Euripides’ Hecuba, 604-608
“You, go and tell the Argives this—
That no one should touch her, that they should keep
The mob from my child. In the great mass of the army
The ungoverned mob, the anarchic fleet,
They’re stronger than fire: and evil is not doing anything evil”
σὺ δ’ ἐλθὲ καὶ σήμηνον ᾿Αργείοις τάδε,
μὴ θιγγάνειν μοι μηδέν’ ἀλλ’ εἴργειν ὄχλον
τῆς παιδός. ἔν τοι μυρίωι στρατεύματι
ἀκόλαστος ὄχλος ναυτική τ’ ἀναρχία
κρείσσων πυρός, κακὸς δ’ ὁ μή τι δρῶν κακόν.
In Euripides’ play, Hecuba laments the sacrifice of her daughter Polyxena at Achilles’ grave. According to some traditions, Achilles’ ghost told the Greeks that they wouldn’t make it safely home unless they sacrificed her. (Her sacrifice mirrors that of Iphigenia at the beginning of the war.)
Although the scene does not appear in Homer, it does appear in early Greek art.
Proclus mentions this in his Chrestomathia (“after burning the city of troy, they sacrificed Polyxena at Achilles’ grave”; ἔπειτα ἐμπρήσαντες τὴν πόλιν / Πολυξένην σφαγιάζουσιν ἐπὶ τὸν τοῦ ᾿Αχιλλέως τάφον). And this is typically assumed as occurring in the lost epic poem the Iliou Persis. Lykophron (line 323) alleges that Achilles had wanted to marry Polyxena—and, according to West (2013, 242-3), it may have been during negotiations with Priam to do so that Paris got the chance to shoot Achilles in the foot.
During the late archaic age and early classical age, Polyxena started appearing in images with Troilus. It is during this episode that some stories may have had Achilles fall in love with her. You know, before or after he killed her brother,
West also notes that the name Polyxena (“many a stranger”, “very hospitable”, “much-hospitality”) may have connoted promiscuity. He refers specifically to Pindar fr. 122:
“Young women of much hospitality, handmaidens
Of Persuasion in wealthy Corinth….
Α′ Πολύξεναι νεάνιδες, ἀμφίπολοι
Πειθοῦς ἐν ἀφνειῷ Κορίνθῳ,
The women addressed in this fragment are allegedly temple prostitutes.