# Representation of $V$ and polynomial rings

Let $$V$$ be a complex vector space. Let $$G$$ be a finite group and let $$\rho: G \rightarrow GL(V)$$ be a representation of $$G$$.

Let $$V^*$$ denote the dual space of $$V$$, and let $$\mathcal{O}(V)$$ denote the algebra of functions $$F: V \rightarrow \mathbb{C}$$ generated by the elements of $$V^*$$. Elements of $$\mathcal{O}(V)$$ are called regular functions.

The dual representation $$\rho^*: G \rightarrow GL(V^*)$$ is given by $$(\rho^{*}(g)f)(v) = f(\rho(g)^{-1}v), \forall g \in G, f \in V^*, v \in V$$

Define the ring of invariant functions to be $$\mathcal{O}(V)^G = {\{f \in \mathcal{O}(V) : gf = f \hspace{2mm} \forall g \in G}\}$$ where $$gf$$ is just shorthand for $$\rho^{*}(g)f$$.

Now, when we restrict to $$V^{*} \subset \mathcal{O}(V)$$, where $$V^{*}$$ has basis $$x_1,...,x_n$$, the regular functions are polynomials in the $$x_i$$ and the action of $$G$$ is given by $$gp(x_1,...,x_n) = p(gx_1,...,gx_n)$$ for some polynomial $$p$$.

$$(1)$$ - I don't understand this last bit? What is the distinction between $$V^{*}$$ and $$\mathcal{O}(V)$$? Is $$V^{*}$$ just the set of $$\mathbb{C}$$-linear sums of $$x_1,...,x_n$$? There is no multiplication between the $$x_i$$ defined? And an element of $$\mathcal{O}(V)$$ is of the form $$p(x_1,...,x_n)$$ for any polynomial $$p$$ with complex coefficients(as multiplication is defined here)?

$$(2)$$ - What does restrict to $$V^{*}$$ mean? Does it mean to restrict the action of $$G$$ to $$V^{*}$$? I thought the action was only defined on $$V^{*}$$ to begin with?

$$(3)$$ - Also, when something like $$\mathbb{C}[V]$$ is written in this context, what does this mean? What does $$G$$ acting on $$\mathbb{C}[V]$$ mean?

For (2), the idea is that the statement is meant to define the action on all of $$\mathcal{O}(V)$$, so the part about restriction to $$V^*$$ seems to be an error (since you are correct that the action was originally just defined on $$V^*$$).
For (3), $$\mathbb{C}[V]$$ refers to the polynomial ring in $$n$$ variables where $$n$$ is the dimension of $$V$$, with the variables identified with a basis of $$V$$. But writing is like that means we don't need to pick a basis, which can have some advantages.
The main idea is that any representation on $$V$$ can be extended to $$\mathbb{C}[V]$$ similarly to what was done for $$\mathcal{O}(V)$$.
• The action was defined on $V^{*}$, so how would the group act on $\mathcal{O}(V)$? Also, how exactly is this representaion extended to $\mathbb{C}[V]$? Mar 31, 2019 at 1:12