Graded vector space

Graded dimension

Let be an -graded vector space. Then has a graded dimension iff for all and it is given by the Formal sum¹ #m/def/linalg

where is a sort of rig of functions. The graded dimension is also called the Hilbert-Poincaré series.

Properties

The following hold when they are well-defined

1. Quotient graded vector space:
2. Direct sum of graded vector spaces:

In addition, if are -graded where is a monoid,

3. Tensor product of graded vector spaces:
4. Shifted graded module: Under the shifting we have

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