You can go here to see my analysis of characters in Black Swan.

One of the first things that struck me in the reading of Black Swan was how Nina’s mother is identified. She’s never actually identified as Nina’s mother – instead, the script just says Erica and describes her as an older woman. I knew from the movie that she was Nina’s mother, but from the script, she could be anyone – guardian, roommate, etc. It actually takes awhile for Nina to refer to her as mom. And it struck me as really odd, considering the relationship that Nina and her mother have (though I thought the relationship was much clearer and tighter in the film version – Nina is much more passive/child-like and her mother much more smothering there). Perhaps this was just accidental, that the screenwriters knew her so well that they just overlooked it, but this was a light bulb moment for me. I realized that even the names/labels you give your characters can affect the tone of your script. Calling her Erica here immediately implies that she and Nina have a distant relationship, which is not at all the case. What if they had called her the more apropos Mommy? It made me realize that even something this minor can have an effect.

I think it's really hard to classify these characters into single narrative functions. Nina is both the protagonist and her own nemesis, as she is both the black swan and the white swan. Because of that, it seems that the other characters play different functions depending on which side of Nina is active at the time. Erica is the loving, if smothering mother, to "good" Nina, and a nemesis to "bad" Nina. Lily is a trickster to good Nina, but an attractor and maybe even a mentor to bad Nina; I would say the same about Thomas. I don't know about Beth -- mentor, perhaps? She never actually helps Nina, but it seems enough that Nina wants to be her. On another note, if this was my script, one thing I would have done would be to combine the Lily and Veronica characters. It seems odd to me that Veronica is Nina's main rival for the Swan Queen role at the beginning, while Lily is the main rival at the end -- like, where does Veronica go? She just disappears. In this respect, I think Lily would be a stronger character (and a stronger trickster) if she had always been Nina's main rival.