Symbolism, Imagery, Allegory
Or, as we've come to think of it, the red sweater of doom.
While the red sweater plays an important role in the story, it doesn't necessarily represent one single thing or quality. Instead, it can be seen as representative of many things. Each reader will probably have a different idea about what the sweater represents.
For example, the sweater could be seen to symbolize the power other people have over Rachel. If we view the classroom as a mini-social sphere, then it becomes obvious that Rachel doesn't have much say on how things go down. Mrs. Price sets the sweater on Rachel's desk, and even though Rachel doesn't want it there, she cannot remove it. She slides it to the corner of her desk and even scoots her chair, but she cannot escape it.
Then, when Mrs. Price tells her to put the sweater on, Rachel cannot argue. Instead, she puts the sweater on, even though it doesn't fit, smells of cottage cheese, and repulses her with the germs that are not hers. The power others hold over you can be both scary and surprising, as we see is the case with Rachel and the nasty old red sweater.
On the Other Hand
The sweater could represent social stigma, or, to quote our favorite sociology textbook, "the disapproval of a person because they do [sic] not fit the required social norms that are given in society" (source). After all, it is "an ugly sweater with red plastic buttons and a collar and sleeves all stretched out like you could use it for a jump rope." It isn't Rachel's sweater, but she notes that "even if it belonged to [her she] wouldn't say so" (8).
We wouldn't say it represents any particular social stigma—such as economic status or gender/ethnicity identity—but more the concept as a whole. When the red sweater is forced upon Rachel, she feels herself separated from the classroom, singled out. No wonder Phyllis Lopez didn't own up to the sweater until nearly lunchtime.
The sweater could represent both of these concepts at the same time, too. One could say that Mrs. Price, as the authority of the class's micro-society, has the ability to decide who should and should not be singled out with social stigma.
Red Letter Day
Everything that is red in the story is associated with two things: embarrassment…and that pesky sweater. Rachel thinks of the thing as "a big red mountain" (13) when it's sitting on her desk, and later she imagines the entire day as a "red balloon" that she wishes would just fly away (22). We wanted to focus on the sweater itself in this section, but one could broaden the discussion to think of red in the short story as playing against its traditional connotations of love or passion, instead symbolizing embarrassment and unease.