In "The Love Song of J. Alfred Prufrock" who/what do the mermaids represent, and why does Prufrock not think that they will sing to him?
The answers to your question lie, in part, directly in the context of the lines you refer to in Eliot's "The Love Song of J. Alfred Prufrock." Here are the lines:
Shall I part my hair behind [to cover a bald spot?]? Do I dare to eat a peach [because he wears dentures?]?
I shall wear white flannel trousers, and walk upon the beach.
I have heard the mermaids singing, each to each.
I do not think that they will sing to me. (lines 122-125)
First, the mermaids will not sing to him because he's old. He is not socially gifted, and his prime, if he ever had one, is long past.
The mermaids can probably be identified by other lines that follow those above:
I have seen them riding seaward on the waves
Combing the white hair of the waves blown back
When the wind blows the water white and black.
We have lingered in the chambers of the sea
By sea-girls wreathed with seaweed red and brown
Till human voices wake us, and we drown. (126-131)
The mermaids are an allusion to the sirens of Ancient Greek myth, who would sing to lure sailors to their underwater caves, then stop singing (breaking their spell), thereby drowning the sailors. The sense here is that the speaker daydreams, and then is awakened from his daydreams by someone talking, since the voices are "human voices." This, of course, connects the mermaids/sirens to the women with, or, I should say, in the room, with Prufrock.
But neither the women, nor the sirens, who are not particularly picky about the men they drown, will sing to Prufrock.
The protagonist of T. S. Eliot's "The Love Song of J. Alfred Prufrock" is a middle-aged man, socially awkward and somewhat shy, who is attracted to younger, beautiful women. He is very conventional and afraid of violating social rules.
The mermaids in the poem represent the unattainable women to whom he is attracted. The fact that they are mermaids, fantastic creatures who inhabit the ocean, emphasizes that they are objects of fantasy rather than realistic aspirations.
I agree with the other educators that the obvious reference is to the Sirens, beautiful and fatal women in Homer's Odyssey who lure men to death with their songs.
The stanzas also strike me as echoing Matthew Arnold's "The Forsaken Merman," a poem that would have been known to Eliot. Although the genders are different, much of the imagery is similar. In particular, Eliot's description of the undersea world evokes the Arnoldian one:
Sand-strewn caverns, cool and deep,
Where the winds are all asleep;
Where the spent lights quiver and gleam . . .
I think that, in some ways, the mermaids Prufrock describes at the end of the poem are also representative of the women at the party. He has said, many times, "In the room the women come and go / Talking of Michelangelo." The women seem to take no notice of him, and they enter and exit the room as though he is not even present. Prufrock seems to idolize and idealize these women—"Arms that are braceleted and white and bare / (But in the lamplight, downed with light brown hair!)"—and he wonders, "Is it perfume from a dress / That makes me so digress?" There is already such a dreamy, detached mood to the poem that the women seem to float like fantasies.
Then Prufrock says, "I should have been a pair of ragged claws / Scuttling across the floors of silent seas," suggesting that he ought to have been a crab, with a hard and insulated shell. Later, when he starts to talk about mermaids, he is likely still thinking of himself as crablike, namely, on the seafloor and alone. Those mermaids, then, would be the women who surround him but do not notice him.