# GIT Lecture 5

This lecture ( GIT5 ) was given on 9/4/09.

### Excerpts

In the lecture, is always a finite group.

A worked example of how to apply Molien’s formula is worked out for the case is the group of symmetries of the cube including those that don’t preserve orientation (so it has order 48). It can be embedded as a subgroup of so in particular it acts on . The result is (after some clever factoring) that

Compare this with the example right before the proof of Molien’s formula in the previous lecture. It follows that is a finitely generated module for a ring of the form where the subscripts denote the degree of the element. In particular the numerator requires the degree 18 peice of the module to be generated by a single element call it . The claim is that there is a relation

with .

Proposition: Let act on an irred. affine var. faithfully. Set and .

- is an integral extension.
- is the field of fractions of
- is a normal extension with Galois group .

proof: take , then . This shows 1. For 2 let . Applying the Renolds operator fixes it so

It can be checked that the far rhs is a ratio of two elements in . Alternatively, let as above. By 1, , so , now is clearly invariant. Now

now the bottom is invariant so its not hard to show the top is invariant too; that is, is invariant, separating each factor into its fixed and non fixed part it follows that is invariant, from which it follows that is invariant. This has the advantage of not using the Renolds operator and so makes no assumption on the characteristic.

Finally, is the splitting field of all polynomials .QED.

### Groups Generated by Reflections

Let be a vector space over latex or characteristic 0. is called a reflection if it is not the identity, has finite order, and a hyperplane that fixes. Let be defined by the linear equations . In some bases can write (for finite dimension n)

Using coordinate then . The next two lectures will be devoted to proving the following theorem:

Thm (reflection theorem): , say . Then is isomorphic to a polynomial algebra iff is generated by reflections.

Rmk: This says will be nonsingular. Further by the above proposition will be a normal and separable (by char K = 0) hence a finite extension, consequently must have the same dimension (i.e. polynomial rings in the same number of variables).

Some preliminaries. For and a reflection. Recall from lecture two that is a reductive group. Dualzing

We get on ring maps

so writing it follows that , so the map

is well defined and lowers the degree of by at least 1. It can also be checked when .

has a natural grading so let .Here is an important lemma that is used more than once

Lemma: Let be homogeneous with . Suppose

and . Then .

comments on proof: its done by induction on the degree of . When it has degree 0 you can divide by it and get the result directly. The key is that when the degree is the operator described above lowers the degree of an element by 1. Assuming the result for , and say it fails for some , then applying any gives the same hypothesis of the lemma except possibly that . But the result should hold given the degree of , since it fails we must not satisfy the hypothesis. The rest of the proof is as in the notes :).

Now begins the proof of the theorem. let be a minimal set of generators for ordered so . Step 1 is to show for each generator there is such that . Assuming this fails there is an all of whose partials are in , but for degree reason can only be made from . Now using the Euler identity you can generate from the generators before it, contradicting minimality.

For the next part in the notes, choosing a relation of minimal degree means in particular, that no factors out of , i.e. are working with something like vs. . Playing around with this idea you can probably justify the all the partials of will be lin. indep.

Plugging in gives , so all its partials are zero. The notes should read ‘ choose i so that ‘ because we want to apply the lemma above. Note that the are invariant because they are polynomials in the generators of the ring of invariants :).

## About this entry

You’re currently reading “GIT Lecture 5,” an entry on Math Meandering

- Published:
- September 9, 2009 / 10:13 am

- Category:
- GIT (course)

- Tags:

## No comments yet

Jump to comment form | comment rss [?] | trackback uri [?]