## § 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.
#### § Example 2: The action of $SL_n(2)$ on $\mathbb C^2$

#### § Binary quantics

#### § References