GIT lecture 7

This lecture ( GIT7 ) was given on 9/11/09

Excerpts

Regarding the example of the alternating group.  Let $f \in k[x_1, ..., x_n]^{A^n} \subset S_n$, then $S_n/A_n = \mathbb{Z}/2$ acts on $f$.  By the group of order 2 can only act on a vector space trivially or by multiplication by $-1$, because are all the irreducibly representations.  Hence all $f$ are semi invariants, from a lemma last time these can be written as a product of $f_O$ for the orbits of the subset of reflections, in this case there is only one orbits, so only one semi invaraint, so the claim about the structure of $k[x_1, ..., x_n]^{A_n}$.

Also in the example of the symmetries of the cube, invariants and semi invariants $h_i(x,y,z)$ were constructed and from these we obtained (using $\gamma \colon \mathbb{C}^2 \to \mathbb{C}^3 \cong Sym^2(\mathbb{C}^*)$ ) to get $f_i(x,y) = h_i(x^2, xy, y^2)$.  and there was a relation $f_{18}^2 + f_{12}f_8^3 + f_{12}^3 = 0$, but plugging everything an multiplying did not seem to give zero.

Consider smooth maps $\phi \colon \mathbb{C}^m \to \mathbb{C}$ with $\phi(0) = 0$.  The problem is to classify germs of such maps.  The map $\phi$ is simple, ($\phi^{-1}(0)$ is simple singularity) if a small nbd of $\phi$ intersects nontrivially with only finitely many orbits under the action $H = \widetilde{Diff} \mathbb{C}^m \times \widetilde{Diff} \mathbb{C}$ (the twiddle means only defined in a small nbd, and Diff stands for diffeomorphisms).  Not really sure what this means….

Finite subgroups of SU(2)

Recall the double cover $\mathbb{Z}/2 \to SU(2) \to SO(3)$.  So for $G \subset SU(2)$ finite, we get an image group $H \subset SO(3)$.  The upshot is the image group is either a rotation group, a dihedral group, or rotations of a polytope.  The vertices of the polytope can be recovered as $P = \{p \in S^2| stab(p) \ne 1\}$.  Its a finite set with an $H$ action, so it can be partitioned into orbits.

$P = P_1 \sqcup ... \sqcup P_d$

The key idea is to count the cardinality of the set $\alpha = \{(h,p)| p \in P, g \in H - 1, gp = p \}$.  This group is a subset of SO(3), so its a rotation hence fixes two points on the sphere, so $\# \alpha = 2(|H| - 1)$.  Setting $h_i = |stab(p_i)|, p_i \in P_i$ then $|H|/h_i$ is the order of $P_i$.  So $\# \alpha = \sum_{i = 1}^d |H|/h_i \cdot (h_i - 1)$.  In particular, we get

$d = 2 + \sum_1^d 1/h_i - 2/h$

Since these things are all integers it can be checked only $d = 2,3$ are possible.  From this the order of $H$ can be determined.  It turns out that $d = 2$ corresponds to rotations, $\mathbb{Z}/n$.  Then $d = 3$ and various different values of the $h_i$ gives dihedral groups and the rotational symmetries of the tetrahedron, the cube (or octahedron), and dodecahedron (or icosahedron).  This first part is essentially the same as what is presented in Artin’s Algebra book.  The notes also contain a table of geometric quotients you get when modding out by these groups.

Lie Algebras & Algebraic Groups

The connection between the description $T_x X = \{ \phi \in (m_x/m_x^2)^*\}$ and $T_x X = \{ d \in Der(O_X \to k)\}$ is given by $\phi \mapsto d_\phi$ defined by $d_\phi(f) = \phi(f - f(x))$ and $d \mapsto \phi_d = d|_{m_x/m_x^2}$.

For $X = G$ an algebraic group, consider all $d \in Der(k[G] \to k[G])$.  Then we can single out the right invariants ones.  By abuse of notation, also denote the backwards map $G \to G$ induced by $d$ by $d$.  Right invariant mean $G \xrightarrow{R_g} \to G \xrightarrow{d} G$ is the same as $G \xrightarrow{d} \to G \xrightarrow{R_g} G$.  And this should be true for all $g$, and the map $R_g$ can be written as the composition $G \times g \to G \times G \xrightarrow{m} G$, which gives a ring homomorphism $k[G] \xrightarrow{\Delta} \to k[G] \otimes k[G] \to k[G] \otimes k[g] \cong k[G]$, so right invariance can be described using the hopf algebra maps, and we the Lie algebra is defined

$\mathfrak{g} = \{d \in Der(k[G] \to k[G])| \Delta \circ d = (d \otimes identity) \circ \Delta \}$.