§ What is a syzygy? (WIP)
§ The ring of invariants
Rotations of We have a group which is acting on a vector space
. This preserves the length, so it preserves the
polynomial . This polynomial is said to be
the invariant polynomial of the group acting on the vector space
. But how do we attach meaning to the symbols ?
Well, we can formulate them as linear operators on the vector space
, and . Then the expression
is a polynomial of linear operators of . So the objects
that are involved in the story are:
We can write any polynomial of the linear operators as polynomials of the
'elementary' projection operators [where is the dimension of ].
So we have that .
Now this ring is said to be the ring of invariants of the action of the
group onto the vector space .
We haven't seen what a syzygy is yet; We'll come to that.
- The vector space of dimension over the field
- The group that acts on the vector space
- The space of linear operators
- The ring of linear operators .
§ Example 2: The action of on
§ Binary quantics