Graded vector space

Graded dimension

Let 𝑉 =⨁𝛼∈𝑆𝑉𝛼 be an 𝑆-graded vector space. Then 𝑉 has a graded dimension iff dim⁡𝑉𝛼 <∞ for all 𝛼 ∈𝑆 and it is given by the Formal sum¹ linalg

dim∗⁡(𝑉;𝑥)=∑𝛼∈𝑆(dim⁡𝑉𝛼)𝑥𝛼∈ℕ𝑆;𝑥

where ℕ𝑆;𝑥 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: dim∗⁡(𝑉/𝑊) =dim∗⁡𝑉 −dim∗⁡𝑊
2. Direct sum of graded vector spaces: dim∗⁡⨁𝑖∈𝐼𝑉𝑖 =∑𝑖∈𝐼dim∗⁡𝑉𝑖

In addition, if 𝑉,𝑊 are 𝔄-graded where (𝔄, +) is a monoid,

3. Tensor product of graded vector spaces: dim∗⁡(𝑉 ⊗𝑊) =(dim∗⁡𝑉)(dim∗⁡𝑊)
4. Shifted graded module: Under the shifting 𝑉𝛼 ↦𝑉𝛼+𝛽 we have (dim∗⁡𝑉)new =𝑥𝛽(dim∗⁡𝑉)old

---

tidy | en | SemBr

Footnotes

1. 1988. Vertex operator algebras and the Monster, §1.10, p. 42 ↩