I am reading about algebraic groups. I don't fully understand the purpose of the coordinate ring, but I feel this is a way of "parametrizing" characters on the group. Here is an example to illustrate:
Consider the algebraic group $G=GL_2$ defined over a field $k$, and $T=GL_1\times GL_1$ inside it. The (algebraic) characters of this torus look like $$ \begin{pmatrix} a & \\ & b \end{pmatrix} \mapsto a^n b^m $$ for integers $n, m$.
On the other hand, the coordinate ring is $$\mathbb{C}[T] = \mathbb{C}[D(X_1) \cap D(X_2)] = \mathbb{C}[D(X_1X_2)] = k[X_1, X_2]_{X_1X_2} = k[X_1^\pm,X_2^\pm]$$ More precisely, a basis for this coordinate ring is given by the monomials $$X_1^n X_2^m$$ for integers $n, m$.
Is this a general fact, i.e.
Is the $\mathbb{C}$-algebra of the characters of a group always isomorphic to its coordinate ring?