Kevin: C[x,y] is naturally Z-graded by deg x = 1, deg y = 1. this induces a Z-grading on the ring of invariants A = C[x,y]^G. If G is not cyclic of odd order, then A is supported in even degrees, i.e. A_n = 0 for n odd. this is the natural grading on A. One often changes the grading by A_n = A_{n/2}; this is called the reduced grading.
There are also fractional gradings: Write C[x,y] = Sym V, where V is a two dim'l vector space. Let R = C[x^2,xy,y^2] = C[u,v,w]/uv-w^2. If we set deg (u,v,w) = (1,1,1), then uv-w^2 is homogeneous. then x and y have degree "1/2".
Incidentally, this is what all the "Spin 1/2" business is about. V is called D_{1/2}. V^{otimes 2j} is called D^j. The Clebsch-Gordan formula tells one how to decompose tensor powers of V. It says D^j otimes D^k = D^{abs j -k} oplus D^{abs {j - k}+1} oplus ... oplus D^{j+k}. see Varadarajan Supersymmetry chapt 1.
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