§ What is a syzygy?
Word comes from greek word for "yoke" . If we have two oxen pulling, we yoke
them together to make it easier for them to pull.
§ 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 what does even mean? well, they are linear function .So is a "polynomial" of these linear functions.
§ How does a group act on polynomials?
- If acts on , how does act on the polynomial functions ?
- In general, if we have a function where acts on and (in our case, acts trivially on ), what is ?
- We define .
- What is the ? We should write. This is like . We want to get.
- If we miss out we get a mess. Let's temporarily define . Consider. But wecan also take this as . This isabsurd as it gives .
We have acting on , it acts transitively, so there's no interesting non-constant
invariants. On the other hand, we can have act on . So if
This action preserves the polynomial , aka the determinant. anything that
ends with an "-ant" tends to be an "invari-ant" (resultant, discriminant)
§ acting on by permuting coordinates.
Polynomials are functions . Symmetric group
acts on polynomials by permuting . What are the invariant
These are the famous elementary symmetric functions. If we think of
- The basic theory of symmetric functions says that every invariant polynomialin is a polynomial in .
§ Proof of elementary theorem
Define an ordering on the monomials; order by lex order.
Define iff either
or or and so on.
Suppose is invariant. Look at the biggest monomial in .
Suppose it is . We subtract:
This kills of the biggest monomial in . If is symmetric,
Then we can order the term we choose such that .
We need this to keep the terms to be positive.
So we have now killed off the largest term of . Keep doing this to kill of completely.
This means that the invariants of acting on are a finitely generated
algebra over . So we have a finite number of generating invariants such that every invariant
can be written as a polynomial of the generating invariants with coefficients in . This is
the first non-trivial example of invariants being finitely generated.
The algebra of invariants is a polynomial ring over . This means that there
are no non-trivial-relations between . This is unusual; usually
the ring of generators will be complicated. This simiplicity tends to happen if is
a reflection group. We haven't seen what a syzygy is yet; We'll come to that.
§ Complicated ring of invariants
Let (even permutations). Consider the polynomial
This is called as the discriminant.
This looks like , , etc. When acts on ,
it either keeps the sign the same or changes the sign. is the subgroup of that
keeps the sign fixed.
What are the invariants of ? It's going to be all the invariants of , ,
plus (because we defined to stabilize ). There are no relations between
. But there are relations between and because
is a symmetric polynomial.
Working this out for ,we get .
When gets larger, we can still express in terms of the symmetric polynomials,
but it's frightfully complicated.
This phenomenon is an example of a Syzygy. For , the ring of invariants is finitely generated
by . There is a non-trivial relation where .
So this ring is not a polynomial ring. This is a first-order Syzygy. Things can get more complicated!
§ Second order Syzygy
Take act on . Let be the generator of . We define
the action as where is the cube root of unity.
We have is invariant if is divisible by , since we will just get .
So the ring is generated by the monomials .
Clearly, these have relations between them. For example:
We have 3 first-order syzygies as written above. Things are more complicated than that. We can
write the syzygies as:
- . So .
- . So .
- . So .
We have in . Let's try to cancel it with the in . So we consider:
So we have non-trivial relations between ! This is a second order syzygy, a sygyzy between syzygies.
We have a ring . We have a map .
This has a nontrivial kernel, and this kernel is spanned by . But this itself
has a kernel . So there's an exact sequence:
In general, we get an invariant ring of linear maps that are invariant under the group action.
We have polynomials that map onto the invariant ring. We have
relationships between the . This gives us a sequence of syzygies. We have many
We can see why a syzygy is called such; The second order sygyzy "yokes" the first order sygyzy. It ties together
the polynomials in the first order syzygy the same way oxen are yoked by a syzygy.
- Is finitely generated as a algebra? Can we find a finite number of generators?
- Is finitely generated (the syzygies as an -MODULE)? To recall the difference, see that is finitely generated as an ALGEBRA by since we can multiply the s. It's not finitely generated as aMODULE as we need to take all powers of : .
- Is this SEQUENCE of sygyzy modulues FINITE?
- Hilbert showed that the answer is YES if is reductive and has characteristic zero. We willdo a special case of finite group.