Sorry, I do not understand what you said . But I can answer the question "are these charachter definitions the same". They do not have to do. I can give you two other references of the word charachter. In number theory a (Dirichlet) charachter is a homomorphism from the finite field to the complex numbers. In linear algebra it means: Let F be a field and G be a group, let f(G,F) be the set of all functions from G to F; a charachter in f(G,F) is a group homomorphism between G and F*. So it is very common in math to use the same word to represent a lot of different stuff.