In Faulkner's "A Rose for Emily," characterization of Emily Grierson is accomplished through the speaker, who represents the townspeople. The town, including Emily, is attempting to survive in the post-Civil War South. Emily would have grown up in the ante-bellum South--before the Civil War devastated the South's economy--part of a fairly wealthy family. Emily, indicative of many people in the South during that time, has trouble letting go of the past.
In fact, saying she has trouble letting go of the past is probably an understatement, as we read after we are told her father dies:
The day after his death all the ladies prepared to call at the house and offer condolence and aid, as is our custom. Miss Emily met them at the door, dressed as usual and with no trace of grief on her face. She told them that her father was not dead. She did that for three days, with the ministers calling on her, and the doctors, trying to persuade her to let them dispose of the body. Just as they were about to resort to law and force, she broke down, and they buried her father quickly.
We see her resistance to change again when the town is offered free mail service, and she "refused to let them fasten the metal numbers above her door and attach a mailbox to it. She would not listen to them."
Of course, the most important example of Emily refusing to let go of the past and trying to hold on is her poisoning of Homer Barron. They take rides in the country and spend time together, but Homer, the narrator says, is not the marrying kind. Emily, apparently, refuses to let him go and poisons him with arsenic in order to keep him with her.
Ironically, Emily may actually believe Homer, as she says she believes of her father, is not dead. If so, her mind truly is a mind that clings to the past.