# Tuesday, August 30

:::{.remark}
Goal: understanding quotients of varieties by general group actions, a basic notion for moduli.
The easiest case: finite groups $G\actson X\in\Aff\Alg\Var\slice k$ for $k=\kbar$.
:::

:::{.remark}
Think of $X \subseteq \AA^n\slice k$ for some $N$, with coordinates $\tv{a_1,\cdots, a_n}$, so $X = V(f_1,\cdots, f_n)$.
Note $\OO_{\AA^n} = \kxn$, and regular functions on $X$ are restricted polynomials, so we get a sequence
\[
R = k[X] \from \kxn \from I = \sqrt{\gens{f_1,\cdots, f_n}}
,\]
so $R\in \kalg^\fg$ without nilpotents -- in fact varieties biject with such algebras.
If $G\actson R$ any ring, one can take **invariants**
\[
R^G \da \ts{r\in R \st g^*(r) = r\,\,\forall g\in G} 
\]
which is a subring and a $k\dash$subalgebra of $R$.
Here $g^*$ is defined in terms of pullbacks of functions:

\begin{tikzcd}
	X && X \\
	\\
	& k
	\arrow["g", from=1-1, to=1-3]
	\arrow["\varphi", from=1-3, to=3-2]
	\arrow["{g^*(\varphi)}"', from=1-1, to=3-2]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJYIl0sWzIsMCwiWCJdLFsxLDIsImsiXSxbMCwxLCJnIl0sWzEsMiwiXFx2YXJwaGkiXSxbMCwyLCJnXiooXFx2YXJwaGkpIiwyXV0=)

:::

:::{.lemma title="?"}
If $\size G\notdivides \characteristic(k)$ then $R^G\in\kalg^{\fg}$.[^later_ask]

[^later_ask]: 
For infinite groups, we'll again ask if $R^G$ is finitely generated -- this will be true when $G$ is a reductive linear algebraic group.

:::

:::{.remark}
There is a $k\dash$linear averaging map
\[
S: R &\to R^G \\
r &\mapsto {1\over \size G} \sum_{g\in G} g^*(r)
,\]
noting that $S$ is *not* a ring morphism.

Let $a\in R$ and consider $p_a(x) \da \prod_{g\in G} (x - g^*(a) )$, a polynomial of degree $n = \size G$ whose coefficients are in the subring $R^G$ and are symmetric polynomials in the $g^*(a)$.
Since $p_a(a) = 0 = a^n + \cdots$, $a^n$ is a linear combination of $1,a,\cdots, a^{n-1}]$ with coefficients in $R^G$ and these symmetric polynomials.
So if $\ts{a_1,\cdots, a_m}$ generate $R$ as a $k\dash$algebra, the images of monomials $S(a_1^{k_1}\cdots a_m^{k_m})$ with $0 \leq k_i \leq n$ generate $R^G$.
If $b\in R^G$, one one hand $b = S(b)$, and on the other hand $b = \sum c_k a_I^{k_I}$ so $S(b) = \sum c_k S(a_I^{k_I})$.
Thus the $c_k$ are in the subring generated by elementary symmetric polynomials in the $g^*(a_i)$.

There is another basis for elementary symmetric polynomials given by **Newton sums**.
Recall

- \( \sigma_1 = \sum x_i \) 
- \( \sigma_2 = \sum x_i x_j \)  
- \( \sigma_n = \prod x_i \) 


The Newton sums are 

- $N_1 = \sum x_i$
- $N_2 = \sum x_i^2$
- $N_n = \sum x_i^n$,

and one can inductively show that one can be written in terms of the other.

The advantage is that the averaging operator commutes with sums, so the $c_k$ like in the subring generated by Newton sums of the $S(a_i^{k_i})$
:::

:::{.theorem title="?"}
Assume $G$ is finite and acts on $X\in\Aff\Alg\Var\slice k$.
There is a bijection
\[
\ts{G\dash\text{orbits on } X} \mapstofrom \ts{\text{Points of an affine variety $Y$ with } k[Y] = k[X]^G}
.\]
Writing $X = \mspec R$ (since we're working with varieties over a field), one can write $Y = \mspec(R^G)$.
There is a quotient map $\pi: X\to Y$ which is universal with respect to maps $G\dash$equivariant maps $\psi: X\to Z$ with $Z$ affine.[^in_fact_affine]
This gives a geometric and a categorical quotient.

[^in_fact_affine]: 
In fact "affine" can be removed here and $Z$ can be replaced by an arbitrary variety.

:::

:::{.proof title="?"}
Since $R^G \injects R$ we obtain a morphism $X \mapsvia{\pi} Y$ of varieties and a pullback $k[Y] \mapsvia{\pi^*} k[X]$.
Given $\phi\in k[Y]$, the pullback $\pi^*( \varphi)$ is constant on $G\dash$orbits.
Given two orbits $O_1, O_2$, one can find an invariant function which is zero on $O_1$ and one on $O_2$.
Any finite subset on a variety is closed.
Delete a point from $O_2$ to get a proper containment of sets $O_1 \union (O_2\smts{p}) \subset O_1 \union O_2$ which are both closed in $X$.
This corresponds to a proper containment of ideals, so pick a function vanishing on the former but not the latter and average.
Thus the regular invariant functions separate orbits, and the images of the $O_i$ in $Y$ are distinct, making $X\to Y$ a geometric quotient.

For the universal property, any $X\to Z$ defines a ring morphism $S\to R$, and $G\dash$equivariance factors this as $S\to R^G \injects R$, thus factoring $X\to Y\to Z$.
:::

:::{.remark}
The right classes of groups to take: geometrically reductive and linearly reductive.[^differences_linred]


Over $\CC$ these coincide, and are e.g. $\GL_n(\CC)$ (trivial center, nontrivial $\pi_1$), the classical semisimples 

- Type A, $\SL_n(\CC)$ (nontrivial center, trivial $\pi_1$),
- Types B and D, $\SO_{n}(\CC)$,
- Type C $\Symp_{2n}(\CC)$, 

along with $(\CC\units)^n$, and their products and extensions, and the exceptional groups $E_6, E_7, E_8, F_4, G_2$.
Here linearly reductive means any finite-dimensional representation decomposes into a sum of irreducible representations.

The most useful for moduli: $\GL_n, \PGL_n, \SL_n$.
Note that $\PGL$ and $\SL$ are almost the same, up to a finite group.

For $\characteristic(k) = p$, the only linearly reductive group on this list is $\GG_m^n$, while "geometrically reductive" includes all of these groups.
Over $\ZZ$, the **split** versions $\GL_n(\ZZ), \Symp_n(\ZZ)$, etc still work.

[^differences_linred]: 
The main difference: linearly reductive is a condition after removing a hyperplane, and geometrically reductive involves replacing a hyperplane with a higher degree hypersurface.

:::

:::{.remark}
Nonsplit groups are e.g. those not isomorphic to $\GL_n(k)$ but become isomorphic over $\kbar$.
Examples: compare $\GG_m$ over $\RR$ and $S^1 = k[x,y]/\gens{x^2+y^2-1}$; these only become isomorphic over $\CC$.
:::

:::{.remark}
Let $S_n\actson \kxn$ by permuting variables, then 
\[
\kxn^{S_n} = k[\sigma_1, \cdots, \sigma_n] = k[N_1, \cdots, N_n]
,\]
generated by elementary symmetric functions or Newton polynomials.
:::

:::{.theorem title="Todd-Shepherd"}
Suppose $G\actson \CC[x_1,\cdots, x_n]$ with $G$ finite and generated by pseudo-reflections.
Then the invariants are again a polynomial ring:
\[
\CC[x_1,\cdots, x_n] ^G \cong \CC[z_1,\cdots, z_n]
.\]
:::

:::{.remark}
More generally, a root lattice $\Lambda$ (e.g. for a Coxeter group) gives rise to a Weyl group $W(\Lambda)$, and one can consider $W\dash$invariant functions.
For example, $W(A_n) = S_{n+1}$.
For a torus, invariant functions are characters.
For a Lie algebra $\lieg$, one can show that the $W\dash$invariants of symmetric functions on the torus, $S(\lieh)^W$, forms a polynomial algebra.
The generators are referred to as the *fundamental weights*.

Coming up next: groups of multiplicative type, infinite groups, and generalizing the above theorem by removing some problematic subsets.
:::