§ What is a syzygy? (WIP)

§ The ring of invariants

Rotations of R3\R^3 We have a group SO(3)SO(3) which is acting on a vector space R3\mathbb R^3. This preserves the length, so it preserves the polynomial x2+y2+z2x^2 + y^2 + z^2. This polynomial x2+y2+z2x^2 + y^2 + z^2 is said to be the invariant polynomial of the group SO(3)SO(3) acting on the vector space R3\mathbb R^3. But how do we attach meaning to the symbols x,y,zx, y, z? Well, we can formulate them as linear operators on the vector space : x,y,z:R3Rx, y, z : \mathbb R^3 \rightarrow \mathbb R x(a,b,c)ax(a, b, c) \equiv a, y(a,b,c)by(a, b, c) \equiv b, and z(a,b,c)cz(a, b, c) \equiv c. Then the expression x2+y2+z2x^2 + y^2 + z^2 is a polynomial of linear operators of VV. So the objects that are involved in the story are:
  • The vector space VV of dimension nn over the field kk
  • The group GG that acts on the vector space VV
  • The space of linear operators VV^*
  • The ring of linear operators RR .
We can write any polynomial of the linear operators as polynomials of the 'elementary' projection operators p1,,pnp_1, \dots, p_n [where nn is the dimension of VV]. So we have that Rk[p1,,pn]R \simeq k[p_1, \dots, p_n]. Now this ring RR is said to be the ring of invariants of the action of the group GG onto the vector space VV. We haven't seen what a syzygy is yet; We'll come to that.

§ Example 2: The action of SLn(2)SL_n(2) on C2\mathbb C^2

§ Binary quantics

§ References