§ 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 we can also take this as . This is absurd 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 polynomial in 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 a MODULE 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 will do a special case of finite group.
§ Is inclusion/exclusion a syzygy?
I feel it is, since each level of the inclusion/exclusion arises as a "yoke" on the previous level.
I wonder how to make this precise.