## § What is a syzygy? (WIP)

#### § The ring of invariants

Rotations of $\R^3$ We have a group $SO(3)$ which is acting on a vector space $\mathbb R^3$. This preserves the length, so it preserves the polynomial $x^2 + y^2 + z^2$. This polynomial $x^2 + y^2 + z^2$ is said to be the invariant polynomial of the group $SO(3)$ acting on the vector space $\mathbb R^3$. But how do we attach meaning to the symbols $x, y, z$? Well, we can formulate them as linear operators on the vector space : $x, y, z : \mathbb R^3 \rightarrow \mathbb R$ $x(a, b, c) \equiv a$, $y(a, b, c) \equiv b$, and $z(a, b, c) \equiv c$. Then the expression $x^2 + y^2 + z^2$ is a polynomial of linear operators of $V$. So the objects that are involved in the story are:
• The vector space $V$ of dimension $n$ over the field $k$
• The group $G$ that acts on the vector space $V$
• The space of linear operators $V^*$
• The ring of linear operators $R$.
We can write any polynomial of the linear operators as polynomials of the 'elementary' projection operators $p_1, \dots, p_n$ [where $n$ is the dimension of $V$]. So we have that $R \simeq k[p_1, \dots, p_n]$. Now this ring $R$ is said to be the ring of invariants of the action of the group $G$ onto the vector space $V$. We haven't seen what a syzygy is yet; We'll come to that.