I love The Scarlet Letter, but I surprised to find out that I did not even know what it was supposed to represent. Its significance seemed to change throughout the book. Is there one central symbol the brooch was supposed to symbolize, or did it have its own symbolic character arc?
The scarlet letter has many meanings in the context of the story.
In the beginning, it represented a sin and crime committed. As the story (and Hester) move on though, it becomes a symbol to be proud of (at least in Hester's mind):
On the breast of her gown, in fine red cloth surrounded with an elaborate embroidery and fantastic flourishes of gold thread, appeared the letter 'A.' It was so artistically done, and with so much fertility and gorgeous luxuriance of fancy, that it had all the effect of a last and fitting decoration to the apparel which she wore; and which was of a splendor in accordance with the taste of the age, but greatly beyond what was allowed by the sumptuary regulations of the colony.
The Scarlet Letter, Chapter II
As time passes, the letter changes in meaning. Some people "refused to interpret the scarlet A by its original signification". The A comes to mean "able" in their minds. (Chapter XII)
Eventually, the letter becomes somewhat sacred:
[it has] the effect of the cross on a nun's bosom. It imparted to the wearer a kind of sacredness, which enabled her to walk securely amid all peril. Had she fallen among thieves, it would have kept her safe.
The Scarlet Letter, Chapter XIII
In short, the letter started as a badge of sin, but due to Hester's actions later it became something, if not to be proud of, not to be ashamed of.